Cross dominance. Its actually not that uncommon.

When I was learning to shoot, I could barely hit the ground with a rifle right handed, but had a "knack" for pistols. Then someone spotted how I was holding a pistol, and I learned to shoot rifles and shotguns left handed. The results include the fastest transition to a secondary in a shoulder or chest holster most people have ever seen.

So I know your pain. Autoloaders and pumps will be your friends, lever actions a little less so, and you'll forever look at all the wonderful bolt guns on the rack and know that they will work but not as well for a fast follow up if needed. And I know that learning to shoot left handed sounds like a pain, but it isn't. And you'll be amazed at how your accuracy improves once you've got the reflexes down.

What you are doing right now just sounds painful and a recipie for chipped teeth and long term damage to your jaw hinge.

As for "training" yourself to be right eye dominant, you can learn to shoot with your right eye, just like you can beat a kid every time he holds a pencil in his left hand (which we used to do), but you can't actually change this. This is a hardwired at the genes neurology issue. I've seen people who were taught to shoot with their non-dominant eye and have done so for years, that when coaxed into trying their dominant eye again were faster to acquire targets, faster to recover, and more precise after a couple hundred rounds, because this is what your brain expects. And for others, such as myself, it is so strong that it is nearly impossible.