I teach tango, which is a social dance. Most dancers there have similar problems, but in many ways there’s an easier way out, because the biggest inhibition that keeps most people from dancing in general is looking stupid. Most asshole tango dancers (or social dancers in general) will just blame the other person if they look bad. I do everything in my power to discourage that behavior.
The whole point of the dancing that I do is that it’s social, and that’s how I get around stage fright. I’m there to enjoy dancing and music and an intimate connection with another human being. When I’m dancing, my focus is on my partner, and I don’t care who’s watching me or why. And because I have a responsibility to help my partner love the dance as much as I do, it’s not about me at all. It’s about the partner. I pour my energy into him or her, and what comes out is a beautiful, wonderful dance.
That probably doesn’t help performance oriented studio dances like the trifecta of ballet, tap, and jazz. But you can use similar techniques. The whole point is that the dance isn’t about you. It’s about the dance.
(I mean, it is about you in a way, but it’s really not. The ego gets in the way, like you’re experiencing now.)
You could try wearing a mask. It helps a lot of people out (and, honestly, is part of what I’m getting at).
Also, tango is entirely improvisational. That makes it have a steep learning curve for beginners, but also helps with that. As @wundayatta says, the improvisation can be freeing, because there aren’t patterns and standards saying what you have to do or should do. Just the dance.
Edit: Also also, another good way to get around the fear of looking stupid is to embrace it. You’re guaranteed to look like a complete idiot at some point. You’re better off playing it off like you meant to do it—indeed, have fun looking foolish. Science bears this out! One cross-cultural experience intrinsic to all human beings is that we actually enjoy looking foolish in front of a crowd after we’ve done it. We actually get chemically rewarded in our brain for it.
People will like you more, too.