I'm sure others will have better answers than I will, but I want to comment that a true internship is supposed to be about helping the intern learn the job/craft/field - not about cheap labor, as some (not necessarily you) forget. With my last intern, for instance, I had her sit in on a paper tech, so she could see what it was like - simply observing, and she was released after half an hour or so, to work on some other project. She didn't need to "do" anything during the paper tech, but I wanted her to see how it progressed. As she was a college student, attending classes during the day, I also much more made sure that she didn't stay as late as the others if at all possible, to accomplish the rest of her homework. Meanwhile, the other ASM(s) were getting paid, so they got more of the paperwork responsibilities.
Also, with an intern even more so than with assistants (though I like to do the same with others), I try to go into more of the "why" I'm doing something, rather than just doing it. There will also likely be more "how can we help you with the next step of your career" discussions, and I try to have more of a grain of salt if they don't "get" something the first...or second...time around.
Yours is a bit different of a situation from many, given that you have no ASM, nor anyone planning to stay for the performance part, so the idea of training them as your replacement seems a good way to have worked part of it.
As for a past "scar" - on the closing interview of one internship, I was told several things that I really wish had been brought up earlier, so I could have shown my ability to adapt/change/learn. In fact, one item I learned during that exit interview is something I've held really dearly and now often mention in the first discussion with any new assistants/interns (having to do with how you present yourself). I guess the short version is, "check in" with each other as needed, especially if you feel you have any issues.
Oh, and since the other departments have been doing this, see if there's anything specific to add. Some theatres have an overall "intern meeting" once a week or whatever, so they can all check in with each other and do more of the professional learning opportunity type things.