That's a great clue, that well water didn't do it.
I'd get a company to come and test your water. Maybe a water filter would fix it. I want to get one as our water has both chlorine and ammonia added, which is terrible for skin, hair and lungs.
The construction company I work for uses Kinetico. The system I need is around $1200, so I don't have it yet. Note that adding a water filter will reduce your water pressure, so if your pressure is too low you won't use it (and have the plumber install a bypass when it is installed so you can bypass it easily).
You could also see if just washing your face in distilled water gives you relief.