How do I tell what way I shoot?

Back in Hunters Ed, we learned our dominant eye, and how to tell it. Im left eye dominant, but a right hand shooter. When I shoot a shotgun, this works fine, since you just point and pull, with little aiming involved. But on a rifle, my right eye can barely see the front post, barely even the target, and my groups are marginal at best. When I shoot left handed, everything is more clear, I can see the post, the target, and then some. I shoot pretty good groups that way. One of my problems, though is that I tilt the sights about 10 degrees, for a weird reason I do not know. So is it possible to be an Ambidextrous shooter, or do you have to be either left handed or right handed?