I'm in the same boat; right handed and severely left-eye dominant. My eye dominance is so pronounced that I physically cannot even partially close my left eye without my right one beginning to close as well. I discovered this problem at a very young age the first time I ever shot a rifle and so I just grew up shooting long guns left handed. I know it doesn't answer your questions for "training" to overcome eye dominance, but simply learning early on to shoot left handed has served me well.
You wouldn't believe the funny looks I get doing transitions when I go from shooting an M4 southpaw to grabbing my sidearm right-handed
