I'm also cross dominant, right handed, left eyed. We discovered this when my dad started to teach me to shoot at the age of 7. I was sitting at the bench with the rifle on sandbags and barely able to keep shots from an open-sighted .22 single-shot on a 2' target at 25 yards. He started watching me instead of the target then stopped me. He moved the gun to the left side of the bench and said "Here, try this." Suddenly, I was keeping them all inside a 4" circle with most of those inside a 2" circle. A trip to the eye doctor taught us that just closing or covering my left eye and having me shoot right handed was out because my right eye was 20/180 (it has since improved to 20/60, but can't be corrected by lenses anymore), while my left was 20/20 (now 20/50, but corrected to 20/20 with my glasses).
I have 2 left-handed bolt-actions, one a Savage 12FLV in .308 Winchester and the other a Remington 788 in .308 Winchester. I have no problems shooting levers, semi-autos and pumps left-handed as long as I can get to the safety, which is why I prefer Mossberg's shotguns.