I am another right handed, left eye dominant person. I haven't shot too many pistols (only one time)but...Shooting pistols I kept both eyes open and twisted my head a bit, have only shot two handed. I am pretty deadly with a S&W model 29 now heh.
Rifles and shotguns, I alternate. When shooting righty, I close my left eye, lefty, I close my right eye usually. Shooting informal clays from a cheapo mechanical trap, I keep both eyes open and hold it left handed, and am pretty good that way.
I believe that being able to use both hands effectively is a worthwhile skill to learn, since should you ever have to use these skills to save your life, shooting lefty when you have to come out the left side of cover will expose less of your body than shooting righty and coming out left. I don't hunt, but I'm sure it could help one in the field as well.
It might feel weird at first, but once you get used to it you are essentially ambidextrous.