Yeah I'm the same. It terrifies me walking down the corridor in school if there's a bunch of people I have to walk past, or going past the grass bit where all the younger kids hang out, because I'm sure they'll stare and laugh at me or something. I had to go shopping on my own the other week and it was awful, I felt so self-conscious walking through town. You feel sort of unprotected if you don't have someone with you, like there's no buffer between you and everyone else! That sounds so weird I know, but it's kinda true...Possibly it's the same sorta thing whereby you don't want to sit and have lunch on your own in the canteen, because people will look at you and think you're a sad loner.
I'm not sure what advice to give really...the idea about listening to music could be good, 'coz if you've got some wicked tunes in your ear on your I-Pod, you're not gonna be concentrating on the people around you so much (then again don't do that if you're walking at night or in an area you don't know, when crossing roads etc!). Or just focus on looking at a building or something when you're walking, not a person.
And I get you about not knowing what to do with your hands! I just shove them in my pocket, or it helps if I've got something to carry like files or books, so I can hold them to my chest - plus if you're walking past a group of people you can act like you have to check something in the file or whatever you're carrying so you have an excuse to keep your eyes down!