I am left handed and right eye dominant. I shoot rifles right handed and pistols/revolvers left handed with my right eye. After 20 years of practice I can shoot pistols using my left eye as good at my right one but sticking with one eye keeps things simple. I can imagine the possibility of making an error during a self defense situation by swithing which eye you shoot with and not committing to the change.
I have a friend who right handed and left eye dominant. He is a police officer and he learned to shoot pistols left handed recently and carries that way now. I still don't why he changed after so many years.