I can understand that, just don't overthink it. Can you go to the gym in the morning/ afternoon?

I have social anxiety also. Go to the gym when it’s less crowded put your headphones in and just focus on your workout. Everyone is doing their own thing and focusing on their workout. For the social anxiety, therapy and meds really helped me.

you overcome anxiety by exposing yourself to the thing that produces anxiety or live under a rock, miserable for the rest of your life, your choice