Guys my heart goes out to you with these struggles. I am a very sensitive girl too.
I have been wondering if it would be helpful to try and learn some stock 'small talk" responses to help you get through the work day. Not sure that I can think of any in response to that comment of your Boss but perhaps others can. Try to roleplay in your mind what you can say when you are interrupted at work so you come across well. I often think over interactions with others after the event and try to work out how I could have dealt with it better next time it comes out. Perhaps watch others in the work place and see if there are things you can copy from what they say.
I think the key is not changing your own personality, but making sure people are comfortable with you, and that your work is not being hidden. I have to say that this really seems harder for you in the US than in Aus. Too much self promotion is a bit frowned on here, and I don't think our education system so relentlessly promotes talkativeness.
Is any of this making sense? I am more introvert than extrovert myself, and in particularly am almost phobic about using the telephone in social settings. But I am bit older than you, and have learnt some skills in those years. I am also ideologically committed to hospitality and being welcoming to people, so I have had lots of practice in getting over my shyness.
Taylor, how do you cope with your clients as a stylist, and your customers at AT. I imagine a level of outgoingness if almost required to do those jobs?