I know I’m coming to this thread late — I was searching for something else but this post intrigued me.
I have no problem myself with handshaking, but occasionally as I’ve turned to someone near me at the sign of peace they have just given a little wave, very deliberately not proffering a hand for me to shake. I have always assumed there was a good reason: a hand injury, an immune disorder where they have to avoid germ exposure, a phobia. I can’t remember this ever happening where the person did not at least smile graciously, and I always smile back, hold up a hand in a sort of wave, and offer the greeting. It’s called the sign of peace, right? So let them engage in it in whatever way preserves their peace!
So having experienced this from the other person’s point of view, I’d say your best bet is to do what these other “refusers” have done. Deliberately refuse to offer your hand for shaking. Hold up a hand in a sort of wave instead. Keep it close to your body, maybe with your hand near your face, so there’s no ambiguity and no overzealous person can grab your hand. And make eye contact, smile, and offer the verbal greeting! If they get their feelings hurt after all this, then I would submit that it is their problem. You could pray for them in that case.
But don’t let Old Scratch convince you to avoid Mass because of your anxiety. When I have suffered anxiety, I pray for Christ to grant me peace. I have learned not to pray for the thing making me anxious to go away, because it often doesn’t… I pray for the anxiety itself to go away. I will pray for you too. But by all means, do not avoid Mass!
One more thought… if some nosy soul gives you a hard time and wants to know why you won’t shake their hand, you can truthfully tell them it’s due to illness and leave it at that. OCD and PTSD are real illnesses that can be very severe for some folks. Nobody would insist on you shaking if you had severe arthritis and shaking caused you pain; psychological pain is no different. Pax Christi tecum.