Lots of great comments here on how social awkwardness fades and changes with age and experience, definitely something that happened to me. Personally I've went from being that shy weirdo to a hyper-social product manager who primarily does presentations, phone-calls, events, coordinates people etc.
My success story was all about a series of customer service jobs in college - foodtruck worker, uber driver, english teacher etc. The social awkwardness wears off when you have to rinse and repeat. And when there is a very good, albeit trivial reason for you to be talking and to being talked to. You learn to create a script in your head, you become comfortable, you try new things and you go off of that script. You learn to fail, you learn to fail fast, and since the interactions are brief, there aren't really any consequences. I basically learned how to not give a fuck.
I would say gym is nice for getting yourself into a certain state of mind, but if you want to make progress in social skill you could probably find that kind of random-stranger-interaction activity. Roles help with this, because people find it easier to talk to someone who is in a specific role and there is a reason for the interaction.
There are two other thoughts I have on this subject:
One is that a big tactic that always works against awkward moments is calling them out. Invariably what makes awkward people awkward is that when they do or say something awkward they will pretend it wasn't so or didn't happen. This is only cringe-inducing for the other party, but they will never say anything because they wouldn't want to offend. You can always take control of a social interaction no matter what happens by immediately confronting that you just did or said something awkward. That then becomes a sort of comedic relief and you're free to change the subject, move on etc. When you realize that it is actually that easy, the stakes of any interaction go way down and you become that version of yourself you want to be.
Second, is that for me learning to deal with social interactions has been a bit like getting into cold water of a lake to swim. Inexperienced swimmers will torment themselves inching in and fretting about getting cold. This is only a waste of energy and an internal struggle. Once you realize that the several minutes of cold are inevitable you just jump in and start moving. It's much better to just get an interaction started off on an awkward leg and to warm up whoever you're talking to, than to remain in that stiff anxious place.
From my perspective today I see social awkwardness as a kind of stage in growing up. It did not seem so at the time, but growing out of that time might just have been inevitable and might have happened no matter what.
That being said, growing up out of social awkwardness does feel like finally entering the club of real adults.