Hi! I've also been struggling with social anxiety for a couple years, so I know what it feels like to struggle reaching out to people or going to events (still working on that one, actually). My tips/general advice would be:
Don't play shy/socially anxious characters. I know, I know, you might think it's easy, since hey, you have social anxiety, you know how it works, right? But personally, what I've found is that it actually compounds all the anxiety I'm already feeling. For example, if I'm in a room with a couple RPers I don't know well, then I might already be on edge from that. Then if my character doesn't know them either, then I also have to think about how they're going to be feeling, which is likely even worse than I am, and basically the two just feed on each other. On the other hand, if I'm playing a character that's okay around people, then that gives me something to focus on, cause while I might be anxious, my character isn't, and I want to stay in character.
Remember that people really do want to RP, and that many are actually really shy too. Yes, that initial message is really hard, but most of the time the person getting it is going to be happy that you're interested in their character. Also, there's nothing wrong with writing something up, then walking away for a bit to calm down before sending it. What I'll often do when I'm struggling with anxiety is write a message and have it ready to send, go play another game for a bit, send it, then go back to that game. It sounds weird, but I've found having the distraction actually helps.
Public events are hard. Like, really hard. My best advice for them is to go with a friend if you can, maybe even a couple friends. That way, at least you know you'll have someone to talk to, and you can sort of ease into meeting up with other people if you feel like it. Also, if you're comfortable with those friends I would suggest being honest with them about what you're feeling. Most people are pretty understanding if you tell them "hey, I'm feeling anxious and might need to take a break/leave early".
That said, practice does help. Anxiety is about focusing on what "could" happen, and I've found having positive experiences to point to is a good counter to that. Though, pick the events you're going to do with this with carefully, since some are both bigger and require more interaction than others. My personal recommendation would be going to a few performances to start, cause that way you can be out around people but without being required to interact too much if you start getting overwhelmed, and they tend to be smaller and less fast-paced than tournaments.
Find a good FC and maybe join a few LSs if you haven't already. Having that constant group of people and the spontaneous RP and FC/LS events that goes along with them is also good practice, and you're more likely to be more comfortable around people you at least somewhat know. Even if it's just in a "oh hey, I've seen your name on tumblr/discord/wherever" sense.
And finally, be kind to yourself. Like I said, I've been dealing with this for a few years now, and while it has gotten better, there are times where I still struggle to attend that event or send that first discord message. There are going to be days when the anxiety wins, and that's okay. The worst thing you can do is to start to beat yourself up over it, cause that's only going to make it harder to try again next time.