Note: This post is about "should" in how people think, in human psychology -- not about "should" in some deeper/broader philosophical sense that might apply to general agents.
I think when people talk about how they "should" do something, there's basically three different types of motivation/shouldness that are meant, and I think noticing this helps make sense of the idea of "getting rid of 'should'" that some people talk about. I'll call the three "external", "internal", and "interalized". (I don't really like this terminology but it's what I've come up with. If you have a suggestion for something better, let me know.)
External here refers to things that are motivated entirely externally -- I don't particularly want to do X, but things will not go well for me if I don't. External "should" has no real moral component to it -- here by "moral" I mean that not in the broad consequentialist sense of what one should do (here by "should" I do mean that in a broader sense rather than a human sense!) but rather that thing that people think of as morality (like for example things that deal with other people's welfare and not one's own, or things being forbidden-or-allowed-or-mandatory).
Internal is the opposite, purely internal -- I feel that X is something that needs to be done and so I am internally motivated to do it. Well, OK -- that description has the problem that it doesn't do enough to distinguish it from the third one, "internalized". I'm hoping the distinction will become clear in a moment when I discuss the third.
Internalized is the nasty one that you want to avoid, the source of scrupulosity, the reason that people talk about "getting rid of 'should'". It's when you take someone else's morality, that you are not allowed to question, and internalize it as binding on yourself. As I said -- this is where scrupulosity comes from; it's not a good thing.
The thing is that if you're not already aware of the distinction it can be hard to describe internal or internalized in a way that couldn't also describe the other. Like, internalized masks itself as internal. Above I said that internal is things that you want to do, that you feel need to be done -- but the person in the grip of scrupulosity would just say, yes, I want to do these things, I feel they need to be done; the obligation is not externally imposed but a result of my own internal desire to do the right thing. Such a person would honestly have trouble recognizing the distinction.
But there is a distinction. The two, well, feel different. Internalized "should" feels, well, bad; it feels like something that oppresses you and gets in your way, even as it's notionally your own internal motivation. Whereas what I'm calling "internal should" doesn't. Another detectable difference, I think, is that internalized should has a certain indirectness to it; it's less "I want to do this thing", as it is "I want to do what is right, and I have concluded that this is what is right". But perhaps that's not the best distinguisher since I guess there are circumstances where internal should can have that indirectness as well.
So when people talk of "getting rid of 'should'", it seems to me they mean "internalized should". Taken literally it doesn't make a lot of sense -- you want to get rid of your motivation? You want to get rid of your notion of right and wrong? But fortunately a person can't actually get rid of their own internal should; but in the attempt to get rid of "should", they can free themselves from their internalized shoulds, from other people's senses of right and wrong that they've tried to incoporate unquestioningly into their own. (External "should" has, as mentioned, no real moral component to it and so isn't relevant here.)
Anyway I think a number of discussions of "should" (again, in the human psychology sense, not in a deeper philosophical sense) make more sense in light of this distinction and the frequent failure to recognize it.