Yes I see now the title was a tad misleading and sorry if my reply seems a bit brusque.
I can really relate to the experience you describe of feeling like everyone is focusing on you and judging you, I have that feeling massively in any public place or social situation I'm in. In my case i tend to feel anxious that people are judging me for my appearance, as I have longer hair than most men. If i have to talk to people I feel anxious that they'll ask me about my life and then judge me for how messed-up and unproductive it is.
I also have that experience of feeling more anxious about doing things in my local area than in somewhere further away. The more anonymous I am the less anxious I feel, the better known I am the more anxious I feel. I always try to keep all relationships at a distance so people can never really get to know me because I feel they'll only judge me and reject me. Even on the internet I avoid making friends or becoming friendly with people because it makes me too anxious.
You mentioned joining a class, I had thought about doing that. In fact there was a class I wanted to join quite recently which I had waited for nearly a year to become available but in the end the anxiety got the better of me and didn't sign up. It's the story of my life really.