One day, one step, one breath at a time. First, everyone has something...not to minimize your situation but rather to say that you are not alone. Get the breakouts under control then go forth! Another idea is look for a shop that has private rooms instead of a big open floor. Also, most shops have privacy screens.
For me, I have really bad cellulite. It's really embarrassing for me anytime I get my lower body worked on. Tell my that I wasn't mortified when an artist tagged me in a photo on IG that showed all of my cellulite and saggy butt for the world to see. I know that the world judges...however, there isn't a damn thing that I can do about my aging body. This is the real world, screw the photoshop version of the world!