I too am left eye dominant and shoot right handed, mainly because my right hand is much better controlling the trigger.
My left eye is also my better eye and I use monovision contacts so my brain is trained to use my left eye for almost everything except reading and close work.
Once I get behind a scope and close my left eye, I my right eye takes over and I hit what I aim at, at least for anything I can see.
If I kept my left eye open, I wouldn't be using the scope because my left eye would have been given control by my brain.
Like Brian, I close my left eye to use a scope.
It may be a handicap, but I don't seem to notice it.
I shoot as well or better than the rest of my buddies who are all competent shooters.
Also like Brian, I use my dominant left eye with open sighted hand guns.