I guess I'm one of the lucky ones -- I'm right handed and my right eye is dominant. As I shoot way more shotgun shells than rifle and pistol shells, I have taught myself to shoot with both eyes open regardless of what gun I am shooting, be it scoped rifle, pistol, or shotgun.
As a "range officer" at our local gun club, I've been involved with helping many shooters with opposite eye dominance.
The simplest way to correct opposite eye dominance for shooting is to put a small (1/4") spot of frosted (Scotch) tape on the dominant eye lens of the shooters glasses.
The spot of tape allows the shooter to use their dominant eye for peripheral vision, but blocks that eye from seeing the gun's front sight, so the other eye will take over to see that sight.