I'm right hand/left eye dominant and shot that way for a long time doing the Navy pistol quals. When I started getting into shooting more seriously, I found one (of many) reasons I didn't do well on B8s was shooting cross-eyed. Like JTS11, I just started closing my left eye and forcing myself to shoot pistol with my right eye. Reinforced with improved scores, shooting with my right eye just became natural 99% of the time.
When I transitioned to a red dot on a pistol, I had to still close my left eye to find the dot and not have parallax error. As I became more accustomed to the dot and where to put the gun after a draw, it got better. Nowadays if I'm shooting close, I'll probably have both eyes open and for distance precision shots (25-50y), I'll close my eye. If I'm practicing B8s at 25y, it will depend on my mood and how my eyes are feeling on what I'll do.
Meanwhile, I always shot right-eyed with a rifle without any real trouble.