I suspect that if somebody had given me this advice when I was a student I would have disregarded it, but, well, this is why wisdom is notoriously impossible to communicate. Wisdom always either sounds glib, banal or irrelevant. Oh well:

Anxiety, aversion and stress diminish with exposure and repetition. 

This is something that, the older I get, the more I wish I had had this tattooed onto my body as a teenager. This is true of not only doing the dishes and laundry, but also vigorous exercise, talking to strangers, changing baby diapers, public speaking in front of crowds, having difficult conversations, and tackling unfamiliar subject matters. All of these are things that always suck, for everyone, the first time, or the first several times. I used to distinctly hate doing all of these things, and to experience a strong aversion to doing them, and to avoid doing them until circumstances forced me. Now they are all things I don't mind doing at all.

There may be "tricks" for metabolizing the anxiety of something like public speaking, but you ultimately don't need tricks. You just need to keep doing the thing until you get used to it. One day you wake up and realize that it's no longer a big deal.

What you really wanted from this answer was something that you could do today to help with your anxiety. The answer, then, is that if you really believe the (true) claim that simply doing the reps will make the anxiety go away, then the meta-anxiety you're feeling now (which is in some sense anxiety about future anxiety) will go away.

My personal approach is to do something you like but have no time for otherwise that can be done at the same time as the chores. I listen to podcasts or audiobooks while doing dishes, for example. My framing is "without chores, I'd never have time for this fun stuff".

I do have this feeling from time to time. Some stuff that's helped me:

• Simplify, simplify, simplify. Do your chores with as little effort as possible, per https://mindingourway.com/half-assing-it-with-everything-youve-got/. Buy stuff that does your work for you. Your goal is to complete your chores, not to do work; work is in service to the goal, not the goal itself.
• Success by default. Make the easiest, lowest-energy, laziest way to do the thing also the right way. Have exactly one clothes hamper for dirty clothes. Buy clothes that don't need ironing. Put decorative items on unused surfaces so you don't put random junk on them.
• Suggested to me by ACX commenters: random rewards. Pick something you like (chocolates, kisses from your wife, etc.) and set up a timer that goes off at a randomly selected interval which entitles you to that reward. The randomness might help your brain to associate chores with better feelings.
• Make it satisfying: Power-washers for big jobs, steam cleaners, compressed air, powered dish brushes. There's a lot of cleaning content on /r/oddlysatisfying for a reason.

This next one's a lot more speculative, feel free to completely disregard. Seriously, I'm suggesting the following because it works for me, but my anxiety response might be way less than yours is and my method might be totally useless for that.

Anyway, it sounds like you have a strong anxiety response when either doing chores or thinking about them. I had a similar issue, and my strategy was flooding, exposure therapy turned up to eleven. Do not recommend.

My real suggestion is, when you feel these racing thoughts and fast breathing, is to stop the chore for a moment and stand there. Don't try to distract yourself, because the focus is, for a moment, now on the anxiety response itself.

For me, anxiety is a series of waves of intense physical symptoms like fast heartbeat and breathlessness. There's definitely a sense of "here it comes!" before each next wave. One thing that helped me was realizing that, while the anxiety response can be quite long-lived, each wave was only 10-15 seconds long. Noticing this gave me strength; if I make it through this wave, I'll have a few minutes to work with the anxiety before the next. This may not be true for you, though.

You say that the endless nature of these chores terrifies you. In this moment of standing still, your goal is to dig a bit deeper, to try to ask "why does the fact that these chores are endless scare me so much?"

Possible answers could be:

• Endless chores scare me because they'll take up a huge chunk of my finite life.
• Endless chores scare me because every time I finish them, I have to do them again soon; all my previous work has amounted to nothing.
• Endless chores scare me because my thoughts race and my breathing speeds up.

Or something else entirely! It might take awhile to get a good answer, and you might have to think through many of them until one seems to fit.

All this might sound very familiar; you say you're taking anxiety medication, but you may also have heard all this through therapy (Cognitive Behavioral Therapy) or meditation. Again, my advice works for me but may not work for you! Please don't feel bad if this doesn't apply!

Regarding a drug-based solution, I recommend Silexan. Scott Alexander wrote about this not that long ago on ACX, and anecdotally, it's been shockingly effective for anxiety issues that arose for me a year ago.

Regarding an entirely psychological, but still clinical solution? Consider starting online therapy with a psychologist/psychiatrist. This has helped me as well- not to the same extent as the Silexan, I think, but it still helped just because it was a dedicated hour per week of thinking about it and voicing those thoughts to someone. Almost every breakthrough came from me just rationally talking it out, but that was 10 times easier in therapy than on my own because I had nothing else to get distracted by or be doing during that period. That, and talking about one's problems to another party has been proven to improve problem-solving effectiveness as compared to thinking them through internally, even if the other party is doing nothing but listening (or even if they're a literal inanimate object like a rubber ducky).

Finally, for my own anecdotal "folk" solution that I completely made up, but which helped me at one point and might be helpful to you as well: try to imagine two selves inside of you in a dialogue, one of them completely calm and rational, and the other embodying whatever your emotions and impulsive thoughts are (so during a panic attack, this would mean picturing one of you who's having the panic attack and another who's calmly thinking about what to do next to resolve the situation). Then, in the dialogue, picture the rational one consoling the emotional one (which is a perfectly rational thing to do for someone who's stressed or anxious).

That last one, I used for a week or two with moderate effectiveness, and then I transitioned from that to standard self-compassion. I'd heard about self-compassion before, but I didn't really get how to actually do it until I'd gotten some practice with that toy model of the rational and emotional selves.

Consider seeing a doctor about the panicky and stressed feelings. They may test you for hormone imbalances or prescribe you antianxiety medication.

I got a lot of value out of How to Manage Your Home Without Losing Your Mind: https://www.aslobcomesclean.com/book/

The high level takeaway is that these tasks become much more manageable when you establish a consistent set of habits and routines around them. There's a lot of very specific step by step advice for creating those habits. It's still the same amount of work, but feels less overwhelming once you have a grasp on how much work it actually is and experience with each task.

I have a similar problem. What worked best for me was simply removing chores from my life.

I realised I was mentally comparing myself to an unrealistic standard of chore-completing perfection (brushing and flossing three times a day, showering/bathing daily, cleaning my apartment every night, doing the dishes after every meal) and feeling perpetually guilty for being a poor adult when I failed to meet my own demands. I realised that leaving an apartment somewhat untidy until I was expecting company was completely fine, that bathing every other day was quite enough hygiene-wise, started using paper plates and plastic cutlery so I wouldn't have to do the dishes ever again (this alone was the greatest relief), etc. So, my suggestion would be to go through your chore list, and see which are being imposed on you by social convention and eliminate as many as possible. Be rationally lazy. Make life work for you. Half-ass it with everything you've got.

The tasks you are talking about are known in caretaking and occupational therapy as Instrumental Activities of Daily Living. Many people do find them difficult to manage independently.

Seeking out occupational therapy may be helpful. Interventions you might pursue with or without their help include skills training (practicing the task to gain confidence with it, or learning easier ways to do the task), modifying your living environment and making use of tools that help you carry them out with less friction, and looking further into training cognitive skills that help with the task. You might also consider screening for any psychiatric conditions that may make this an additional challenge and look into any useful treatments or management associated with anything you find.

Not a complete answer, but something that helps me, that hasn't been mentioned often, is letting yourself do the task incompletely.

I don't have to fold all the laundry, I can just fold one or three things.  I don't have to wash all the dishes, I can just wash one more than I actually need to eat right now.  I don't have to pick up all the trash laying around, just gather a couple things into an empty bag of chips.

It doesn't mean anything, I'm not committing to anything, I'm just doing one meaningless thing.  And I find that helps.

Some things that help me:

• Having a proper task tracking system with due dates (I use rememberthemilk but it's not necessarily the best one, just what I'm used to.) Once you know that all your tasks are accounted for, you don't need to stress about forgetting something. You should be "stateless", your brain occupied by whatever you are doing at the moment, and not by fretting about all the other stuff you need to do. (stolen from getting things done.)
• Have a dedicated "chores day" once a week when you do all the chores that (i) don't need to be done more frequently and (ii) don't have any other specific time they need to be done. You implement it by assigning every new chore to the next chores day in your task tracker. If, during chores day, you don't finish all of the chores, then move those that suffer delay to next week and the urgent ones to tomorrow. This system makes all non-chore-days both more relaxed and more productive.

The perspective in this post has been quite helpful to me for dealing with this and might work as a new thought paradigm if viewing these things as obligations is part of what is causing your stress. If the never-ending nature of the tasks itself is an issue, then perhaps this post can help with that. 

Something I've found helpful for similar issues is to change my mindset from "I need to x" or (worse) "I should do x" to "I want to do x", even if the reason is because the consequences of not doing it seem very bad. Trying hard to reframe things as a desire of mine, rather than an obligation has been very good for me (when I can do it, which is not always).

If there are too many such tasks, maybe its time to find a way of life which removes most of them:

• Hire a cleaner
• Order food from restaurants or eat outside. 

There's a cluster of things around non-coercion, parts work, and coherence that might be helpful here.

It could be good to start with something like focusing, which can help you find the meaning behind the stress, and may give you more information about how to address it moving forward.

Maybe a shit answer but becoming great at meditation allows you to be with emotions without trouble. Chores then become an oppurtunity to deepen your awareness and practice (more long term but you cut the root of the tree)

Something I find useful for chores is making rules, not individual decisions. (I daresay there’s a connection with TDT, although I learned this before TDT was ever invented.) For example, “all dishes will be washed before I leave the house or go to bed at night”.

Cognitive behavioral therapy for what appears to be fairly severe underlying anxiety?

Rebt in particular might apply as you seem to overwhelm yourself with the thought of the thing more than the thing.

Undiagnosed ADD comes to mind as "existential crisis doing chores" comes up a lot to describe it when I talk to adults.

Unified mindfulness would also be a suggestion, you can use the opportunity of the hated chores to wire up a more peaceful sensory experience and relationship to your body and mind.

Once you reach a certain age, you may find that these little tasks and chores become therapeutic. They offer a chance to step away from your desk and do something with your body. This isn't really a big deal in your teens and 20s, when your body can take all sorts of punishment and keep going, but once you pass the 30 mark, you may find that a little light exercise and movement feels really good if you've been sitting at a desk or staring at a screen all day.

Also, studies have shown that exercise actually improves mental acuity. I don't remember the exact details, but I recall one study that split students into two groups: one that studied for about 1 hour, and one that studied for about 45 minutes and exercised for 15. The second group (the one that studied less and exercised more) actually performed better on academic and memory tests.

So my advice is: don't think of chores as wasted time, but as opportunities to recharge your mind and body and actually accelerate your mental acuity. You might not think of folding laundry as exercise, but compared to sitting at a desk, it is.