If I'm allowed to jump that late in the discussion (I was offline for a while in order to get my endorphin depot filled up on the steepest snow powder slopes I could find in the Alps - skiing is second to sex only imho)
But to the topic of approach anxiety (AA) I have 3 proposals for you that helped me:
1st, do this training always when you walk around: Give eye contact (EC) to everyone passing by. To any stranger, male and female. Smile slightly, not in a way that is implying something, but in a way that you are not threatening. Watch your reaction and the reactions of the other ones. Hold that eye contact so long that is uncomfortable to you (but don't stare, 5 sec are fine, but if you have AA you most likely bail out earlier). If you break the EC do it always sidewards. Looking down is a communication of submission. Watch out for this and learn to avoid it. Learn to identify female EC (
https://markmanson.net/the-levels-of-eye-contact/amp) - once you are proficient in that you see who wants to be approached. That makes things much much easier. See "warm approach"
2nd: Use any chance to start chatting with anybody. Just any non-gaming everyday comments, waitresses, shop assistants, guys and gals which whom you queues up somewhere and so on. Simply to train this muscles. I once spend a nice night with a woman which I get aquainted by chatting with her in the queue of a club. Make the starting comments easy and lightweight, so that is just small talk. Once you get better you can become more risky. This is again a training issue.
3rd, this one is a bit tricky, because it contains the usage of potentially addictive substances. For me it worked great, but you need to have a high discipline and be able to control the usage. I give some background. AA is to some extent a learned behavior. Something in the past gave you (and me) bad experiences. You (and I) need to "unlearn" the AA by having some good experiences. Once you have that new good experiences the AA is gone. The problem is that the AA itself prohibit the new good experiences, it's a circulus vitiousus. For me the following worked well: for a couple of events (read club nights) I used Phenibut and/or Mitrogynin (check if it is legal in your jurisdiction). Both substances lower the level of AA without the impairing effects of alcohol - but both are potentially addictive. By using them very controlled (max. 2 time per week) the addiction can be avoided. Maybe make a schedule that limits the total number of usages. For me it was a temporary fix only. After 5 to 10 good nights I have so many fresh good experiences that give me the "I! Can! Do! It!" attitude, I fear no more and I don't need it anymore.