I had the same "problem"
If you shoot pistols (IDPA, IPSC, etc), it's not a problem, it's a great advantage. That is, once you learn to shoot with both eyes open. It will feel strange at first but after a few thousand rounds, it will become automatic. What this means is that you not only aquire straight on targets faster, but you can shoot to either the left or right by moving just your eyes to the side without having to center your head on the flanking target. This is a good thing for speed. Heck, I even began shooting Bullseye with both eyes open and my scores improved, though I can say for sure that's the reason why.
With rifles, it kinda sucks because it will lead to 'scope eye' quicker, but that's really only a problem if you're varmint hunter or shoot a lot through a scope. You can always use a left hand stock; you'll get used to a left hand trigger pull. I'm thinking about having a custom rifle built, a 6mm prairie dog gun. I may have it done with a left hand stock (cheek piece, palm swell) but with a right-handed bolt. Cool, huh?
With shotguns, a lot of guys like us learn to shoot left handed. Others will learn to shoot with both eyes open, just like with pistols. What has worked for me is to keep both eyes open as I'm watching the bird and as I approach with my point, I close my left eye, swing through and smack the trigger. It's what works for me. I'm not a world class shooter here though I did recently get my first 25 in a row shooting skeet!