Connection not Perfection
Socialising is about having a good time. I think that if you're not enjoying yourself then you're being anxious. Getting over social anxiety requires that you turn socialising into a joyful activity, be true to yourself, stop taking things so seriously, and always be on the lookout for a way to spin a boring conversation into something interesting and funny.
I always thought that conversations had to follow a logical flow. That I was doing things "wrong" if I introduced a random topic or said something that didn't directly follow from the previous thing that was said. In reality, nobody cares or even notices. How many times have you been in a conversation where someone has stopped to say "hold on, what you just said was not linked to conversation topic"? Probably never, right? People are actually glad if you change the conversation topic into one that is funnier or more engaging.
Maybe I had just spent too much time on computers. Each statement in a computer program has to logically follow on from the next, and each line has to be correct. But conversations aren't like that. Which is a good thing, because I wouldn't want to model human interactions from a machine that can't experience joy.
Starting conversations with new people is about creating a connection. You do that by being true to yourself, saying the first things that come to your mind, not by trying to find the perfect thing to say. Perfection can be so paralysing that you over-obsess so much over finding the perfect thing to say that you either say nothing at all, or hesitate so much that you turn the interaction awkward before even speaking. Unnaturalness is the worst thing you can bring to a conversation. If you want connection, then drop perfection.