I am right handed but can't use my right eye much at all, near or far. I grew up playing with toy guns shooting right handed, so when I bought my first gun, a bolt action with a scope for right handed shooters no less, I could not shoot right handed. It took me some time to get used to shooting left handed but I have gotten so used to it that is actually feels wierd to hold a gun right handed.
My opinion, for what it's worth, is that you may be able to get away with a red dot or both eyes open, but you should keep your shooting consistant as possible and just switch to left handed shooting. That goes for rifle (scoped, red dot, iron sights), shotgun and handgun. That way, if you pick up a scoped gun, you dont have to worry that you wont feel comfortable holding it left handed. It won't take as long as you think to switch and it will pay off in the long run. You will just need to practice until you feel comfortable.
The only guns that I can't shoot comfortably are bolt action guns (in right handed configuration) and 1911 pistols without ambi safety. Semi auto rifles (like my AR, M1a and Garand) require no adjustments. I don't really even think about buying the Stag left handed because it's just not a big deal to shoot the AR left handed.
Good luck.