I, too, am right handed left eye dominant. I cant my head when I shoot to align my left eye with my right hand. Lately, I started shooting with both eyes open, although I still squint my right eye on farther targets, I'm able to hit where I'm aiming.
On shooting left handed with a rifle, have you tried doing it right handed with your right eye on the sights? I do it and find no problems.
Guess I'm weird in some way.
Ninja45