When my EC was at its worse, I felt the best way to deal with it psychologically was not to allow it to define who I was. You are not a broken shell of person, you're the same person just with a skin condition. If you feel uncomfortable around people and that people are staring at you, don't take it personally. Your lips might be a part of your face but they are not your identity, you lose the battle if you give them the power to dictate your mood. Separate yourself from them mentally. Treat them like a Science experiment, and yourself a scientist who will do everything in their power to study, research, seek treatment, and eventually overcome.
It's very difficult to overcome this mental hurdle, but if you can do it than you will have a clearer head and be in a healthier, happier state while you are healing. So there is nothing to lose and everything to gain.