Fake it til you make it.
I'm not even joking. If you work on pretending to be calm and socially friendly, you'll eventually become it. You won't necessarily fool someone that's familiar with social anxiety, but that's not the point of it. The point is to build up habits of behavior over time that allow you to function with the anxiety, in a way that guides your mind to being confident with its skills at navigating social situations.
Practice smiling, practice greetings, practice your body language. Just being able to let yourself smile when you're stressed is a huge help. It literally changes the way your brain is processing events around you in the same way that laughter elevates the mood even when it's artificial at first.
It's like breathing exercises. Since our brain brain/body connection is a two way street, if we control the things that are changed by a stress state, we can control the stress itself. That's why deep, controlled breathing eventually calms the mind and shuts down adrenaline dumps. We get stressed, we breathe shallow and fast. Change that breathing pattern, and you essentially create feedback that changes the mind to the state usually present when breathing in the new pattern.
And yes, that works in reverse; intentionally breathing the way you do under stress generates a stress response in other parts of the body. But the good thing is that a neutral state is the default. It's easier to return to a calm but aware state from a stress state than it is to ramp up into a stress state from calm when there's no external threat.