Thanks for the help everyone. I have taken theory courses, although I've forgotten a lot of it, and while I'm quite confident I /could/ analyze a piece and figure out the chord structure, I haven't actually done that with any large piece I've played. I think you're probably right that that's something I should work on so that notes don't feel "random". I do always think about the notes in the key I'm playing in, though. I can visualize the piano with the notes in the key sorta... popping out from the piano. I find that thinking about that does help a lot.
As far as the physicality of being calm goes, that's a big struggle for me and has been the primary discussion of my lessons for the past 3 and a half years with this teacher. And I find, I'm not really playing pieces any more difficult than I played 3 years ago, but I play them much better, with less stress, with less effort, and with less time to learn them. So, I think, if I put in as much time to a piece now as I put into pieces in high school, I'd be able to play stuff at least a step more difficult than what I've currently been playing. Something really clicked last year when playing Brahms' op. 118 where things felt so much easier than they had before. But there was still some mental stress.
I think bernadette's suggestion is a great one. I have noticed that too. Actually, I've noticed that in the occasional week where I will only practice once between my lessons, I actually make a ton of progress. (Obviously, that doesn't work if I do it every week, but...) It seems like spending time away from a piece helps make it easier.
Thanks for all the suggestions, I will work on it.
David