Good news and bad news, it would seem.
First and foremost, it's a good thing that it gets worse in social or stressful situations.
At least it's not something physical going on, and it's caused by psychological issues that can be dealt with.
I had that problem for a while playing guitar whenever I'd play in front of a crowd. The things I did a million times on my own became botched songs that disappointed me because I'd start to shake.
I won't go into what changed my brain to take much of my stress away, as it's much too long for here and probably not suitable for this forum anyway, but the point I would make and try to relate between us is that it wasn't the tremor causing the stress, the stress caused the tremor.
Kind of frustrating, really, because it gets into a vicious cycle that makes itself worse. You've got to find some way to alleviate some of your stress, and to make yourself more comfortable in those situations. One thing I like to do is do simple hand tricks, things like flipping pens in a circle, or rolling a quarter on my knuckles.
It takes dexterity and hand-eye coordination. The trick is, do it in front of people all the time. I do it in class, during guest lectures, pretty much any time i can without getting myself in trouble. I used to solve a rubik's cube the same way, always trying to do it in front of people. My performance anxiety (i'm hesitant to call it that due to obvious negative associations...) extended beyond shredding on a guitar in front of a crowd, and was a deeper issue of feelings of inadequacy or somehow thinking I needed to be more than I was or that I wasn't acceptable.
Get yourself doing things that make you uncomfortable until you realize with the little stuff that it doesn't matter if you mess up once in a while, and everyone makes mistakes. We're old enough that the majority of people don't mock you for simple mistakes, and especially in residency I would hope that those times would be your best opportunities to learn.
See a guy do something right, and it makes you think he knows how to do that. See a guy do something wrong, and you'll find out his willingness to learn, his ability to adapt, his openness to criticism and respect for academic authority. You have more opportunity to be impressive after you mess up than after you do something right.
If these issues are severe, you might look at getting some counseling, or a psychiatrist if you're looking more to get medicated (but I don't know hospital views on that).
Don't let it get you down. I face a lot of the same awkward social feelings and stress, and it makes me perform poorly if i let it get to me. I find that when I'm the top dog (think the difference in how you feel when you start a job as opposed to when you've been there for years and you're teaching a new guy), I'm not nervous or worried what they think because at that point, I know I'm awesome and humble ;-). The more you feel like you belong to the club, the less stressed you'll feel. At least that's how it is with me.
Good luck. I would love to see some resident/attending chime in here.