What I'm going to say is something that's probably incredibly hard to do. But it's helped me in the past.
People staring expect a reaction when you see it. And if you're feeling really insecure and, well, very uneasy with yourself when it happens, the reaction you give may well be just the one they're hoping for. A sort of cringe, almost apologetic not-knowing-what-to-do-with-yourself reaction. Which is entirely understandable. Often it hurts. A lot.
I get from your post that you don't feel confident either way at the moment, sweetie. And feel like you have to hide yourself away. This, perhaps just as much as anything physical about you that you're not happy with, is what adds to the experience. You feel self-conscious so you're hyper aware of everything around you. Every single glance, turn of a head in your direction, any sounds... anything. I've been there.
What I would recommend is, rather like Lady_Oracle suggests, try to own yourself. To practice an air of being confident. It's the hardest thing in the world when you don't feel it, I know. But sometimes you can act confident even when you don't feel it. When you're out and about, keep the thought in your mind that you actually like who you are. Remember why you're doing everything, and what the end result will be, and that in just a relatively short space of time, you'll have something which makes you feel even better about yourself.
If people stare at you, meet their gaze and flash them a huge smile. Really, even though you might feel like you wish the earth would open up beneath you, you would be surprised how often the stares melt away when they don't have the desired effect on you. And even moreso if you stare back. People are only comfortable if they feel like they have some sort of power over you. Turn that around and you take that power back for yourself.
Sweetie, you don't deserve to make yourself feel like crap for a majority of people you don't even know or will never see again. *hugs* I know it's a hard thing to get past. But by owning yourself and your own world, you make it so other people cannot.