I'm right handed and left eye dominant. I learned to shoot at camp when I was about 11 years old. The first thing the instructor did was have us figure out if we were right or left eye dominant, and then that's how we shot. I can count on one hand the number of times I've shot a rifle right handed and it just feels completely wrong. Frankly, I don't know how you guys do it.
Honestly, I see it as an advantage rather than a disability. I'm using my dominant/stronger arm to support the rifle. Shooting semi-auto I can see if brass gets caught in the chamber of fails to eject properly without having to readjust the rifle.
Bolt guns are harder, but I recently bought a few left handed models and problem solved.