I'm right-handed and left-eye dominant. I shoot long guns and bows left-handed. I shoot handguns right-handed with my left eye. I've had a few interesting conversations at the range with firearms instructors about my goofy pistol stance! Shooting with both eyes open is best, IMHO, but it took a while for me to get used to it.
I also tried several things (eye patches, etc) suggested by local experts to "cure" my problem. Nothing worked for me. I decided to practice what came naturally to me and I do OK. Good luck.