I, too, am/was left-eye dominant and still am right handed. I only learned about the eye dominance thing about 2-3 years ago. It has never caused me any issues when shooting rifles. I've always closed my left eye until about a year ago when I started to keep both eyes open after getting an RDS for my rifle. Since I can only see the dot with my right eye I didn't think it would be an issue and it hasn't been.
When I first learned about my eye dominance issue I also figured out that I was lining up my handgun shots with my left eye. It was so automatic that I didn't even realize I was doing it. So, I began concentrating on using my right eye for shooting handguns for a year and it became much easier and accuracy increased. Then, last year, I began focusing on shooting handguns with both eyes open, but continuing to use the right eye for sighting. Accuracy has stayed about the same to slightly better, but speed has increased.
Then about two months ago my BIL came with me to the range to have some fun and started shooting my rifles with both eyes open on iron sights and was doing really well before I even realized I had both eyes open the whole time. COOL! I do dry fire exercises all the time and draw and acquire the sights without firing during the day and at night in the dark. I also have night sights.
Now, when I do the triangle thing with my hands, instead of coming to my left eye as before, they come straight in to my nose. Maybe another year or so and they'll come to my right eye.
Just my 2 cents.
I don't practice that enough.