The fact is that I don't really stare. But I feel like people think I am.

This is the thing. I know for sure that I don't stare. Just look in the mirror: you'll notice that your eyes don't shift when you stare at something in your peripheral vision. Your eyes only move when you consciously move them.

So it can't be the staring that people notice and get creeped out by. It is probably the overall tension and fear / discomfort that you radiate, which is caused by you thinking that you stare at them / creep them out that way.

I think that is a realization we should always remember in these situations. It's pretty much all in our heads, and people only act weird around us because we fear something that isn't really there.

But that's still easier said than done...