Guys I have some ideas as I dealt with this too, but I have always been very outgoing:
MOST IMPORTANT is that I can control it, I learned how, and here are the steps:
it is a confidence issue first and foremost, for me at least. I found that i had too much judgement towards me and that made me unsure on how to behave. When you feel that gut reaction of "are you sure of what you are saying?!" every moment - that is just fear, fear that others will judge you on what you say - well, you can learn to think better to be assure in what you say, or just roll with it, i suggest the former btw.
1. distracting yourself. i've found it helps immensely to be in your own world and think of stuff you need to do. You ARE coming across one way or another, be it a shy person/ confident/ etc, so which way works for you is the way you have to OWN it, which is 2:
2. is simply to say "look, as long as I am not hurting people (and you aren't with staring, you are simply making people think about why you are looking and make THEMSELVES nervous, but they aren't thinking ABOUT YOU. If you walked in the busy street, you wouldnt have this issue after 10 minutes. Also realize if you are freaking out inside people can't SEE inside you, they can only look at you, so "keep it cool" and remember 3, which is most important.
3. is to genuinely wish people well and hope they do the same for you. Notice the highlighted word. If you deal with too many judgmental characters (hell, we all have) - realize this is just THEIR way to deal with life, it isn't necessarily the right one. I've met some smart people who like to judge, but I remember (and sometimes have to remind) them that while they might be correct in one area, they too have areas which they are not doing too well in - so KEEP THINGS IN PERSPECTIVE. Listen to people to excel at MANY things, love, wealth, health - balanced people. Not just those who hold something like money or status over others.