I think it might be helpful to really drill into what you want vs what you're experiencing. You state you have a desire to grow socially, but your attempts to do so have left you feeling symptoms of burnout.
More information about what you feel is expected of you, socially, at work, and what the specific triggers for your negative emotional reactions are would be useful to identify strategies to ameliorate those responses.
Doing some real specious armchair psychoanalysis here, but you're statement that you do not want to be somewhere where you might be recognized indicates to me, specious armchair psychologist extraordinare, that you perhaps have some self-esteem issues which are going to be a significant impediment to socializing in any context, let alone work. I'm casting aspersions from within my glass house here, but in the worst troughs of my depression, I rationalize self-isolation as a protective measure so that I don't have to converse with anyone about my life, since I'm not proud of anything I've done in those moments. It's only when I get myself back into a headspace where I have things in my life that I'm excited about and want to share with people that socialization begins to look attractive again. If any of that rings true with you, you might recalibrate your focus from trying to force yourself to enjoy your professional social life and instead focus on the thing that's actually holding you back from making that a reality.
Good luck, and I hope you find a solution.