Don't worry about people seeing your eyes dilated. Traditionally, dilated eyes were associated with good looks or beauty and in 1800's ladies would put dilating drops in their eyes to look more beautiful. Helps if you are a woman, particularly, I guess. I guess I've looked at hundreds of thousands of people in my time and never had any concern or notice about what the other person's pupils looked like. As far as people seeing you being anxious all the time - people see me that way frequently but I try not to show it. It's not always easy. Meditation or yoga, deep breathing, lexapro, could possibly help you to get into that zone - where you don't outwardly show the stress as much and maybe that could transfer to some degree inwardly to reduce your stress a little. In your case, you feel that you looked stressed and that tends to make you feel more stressed. And that's from an ophthalmologist - so take it for what it worth. I wish you well.
I know this would be really hard for someone with social anxiety to do..but maybe that's why you should do it. I think you would benefit a lot from an introductory acting class. In an introductory acting class there is normally a lot of emphasis on learning to relax and not being afraid of looking silly. I think it would help you build a lot of confidence. Normally people are too self-conscious about themselves to notice what others look like.
I'm sure there are also lots of therapy techniques to help you as well, so maybe you should try to see a therapist. I've also heard that anti-anxiety drugs can do wonders.