I'm right handed and left eye dominant too. I've found that closing my right eye works just fine for handgun shooting but with a rifle I always shoot right handed and close the left eye.

The funny part is that I have often caught myself closing my left eye with a handgun under the stress of fast shooting such as in a combat pistol match with no loss of accuracy. I know this shouldn't work but for me it does. I've had people tell me to shoot a pistol with both eyes open. Tried it and it doesn't work for me. It's got to be one or the other and doesn't seem to matter which.

For bullseye pistol shooting it's left eye open always.

Beats me.