The social anxiety "paradox," if you will, I totally get. And not even from a musical performance standpoint.
As part of my work in public ed, I've been presenting at conferences more and more frequently over the last few years. Put me in front of a classroom of 50+ fellow educators who are sitting there to learn about whatever topic I'm speaking on, and I'm on, man. It's a performance, as you all know. Just that I have a laptop and a USB laser-pointer slide show button remote control thingie instead of a bass guitar. The feedback I've gotten on my presentations have been stellar, talking about my energy, my humor, my ability to translate complex technical concepts into human language, and how I can connect to my audience so that they can actually absorb what I'm presenting.
But then shift me over to the hotel bar later that night where I'm sitting by myself to get a drink and a bite to eat after the conference is done and I am completely shut down, hesitant to strike up a conversation with someone who sits next to me. That introversion rears its head as soon as I'm done with my speaking obligation; I read a succinct phrase once that describes this -- the "extroverted introvert."