I know of a little technique you can use to help with the fear that people are looking at you. The next time you are in a public place, try slowly looking around you at other people; not slowly meaning in "slow motion", but just look around a little more slowly than usual. If you look at other people around you fast, by moving your eyes fast and/or turning your head really fast, it tends to cause you even more anxiety in the moment. Instead, try to slowly look around at people nearby you in a public place to see if they are looking or staring at you. This takes time and practice in order to reduce your anxiety and to prove to yourself that people aren't really staring, but eventually if you keep working on this and practicing it in public, you'll eventually see that people aren't really staring or looking at you and judging.
If it's too challenging to do this at first, ask a close friend or family member to go with you somewhere and do this in public, so that you have their support and you don't feel as comfortable as you would be in a public place by yourself.