I also shoot right handed but am left eye dominant but I disagree with the others that it is difficult to train your self. The approach you are taking is exactly what I did. I'd draw back and get set with my left eye closed and once the pin was where or near where it needed to be I would open my left eye. If you keep at it you will prevail. Now when I draw both eyes are open and I have no issues getting set on my right eye. On occasion I'll notice that something is out of place and by blinking my left eye my right locks on. Keep shooting they way you are and I'm sure it will work out for you.

I've been shooting quite a few years and I did't train myself overnight. When you draw breifly think about your anchors and if needed quickly close your left eye to get your berrings. I learned as a result of not being able to see a deer well in lowlight just looking thru my peep. By opening up your left eye you'll be able to see the whole picture.

Good luck and keep at it.