I know several people with social anxiety and it looks different on each. For one, she can't make outbound phone calls, Another can't make outbound emails. For both, when they must do these things, it's overly formal and stilted. It doesn't sound or read naturally. It's like they become someone else and they write or say what they think they are supposed to say and not just being authentic or being themselves.
One of these has the "white coat" anxiety when going to the doctor, too. She also won't go to a new store alone. She needs someone else to go with her so she doesn't have to "look stupid" about random things, like where do you pay? Where is the line? Which door do you go in? For her, it's all about not wanting to look foolish.
Another doesn't want the unexpected visitor, the unexpected encounter, the change in plans, the unexpected text. She's also a hoarder.
A couple of them won't answer the land line phone or answer the doorbell. A couple hate do deal with repair people who go to the house to fix things. They don't want the social encounter.
From what I can tell, it's a lack of confidence, worry about what people think, fear of being imperfect or vulnerable, trying to maintain some kind of image that isn't the true them, fear of losing control (for the random meetings in stores it is about saying the wrong thing or not being able to get away quickly - fear it will be an hour long chat in the aisle of a store because they think the other person has to bring the conversation to a close, because they feel like it's rude for them to do it - but yet they don't think the other person rude when they say, "Well it was nice chatting with you - gotta run.")
I always thought this is what Brene Brown's books are for.