Being cross dominant is pretty common. Probably about 20% of the population has a master eye that is NOT on the same side as their dominant hand.
(Do a search on "the firing line" and some of the other online shooting forums. This is a COMMON issue of discussion!)
I am right handed and have a left master eye.
When shooting a handgun, I shoot right handed, shift my head position slightly, and use the left eye to sight with. This allows me to keep both eyes open. (Since I'm a cop and a handgun is my primary weapon, I felt it was an appropriate adjustment to make. I first learned to shoot with a BB rifle and learned to sight with my right eye, and that worked fine)
When shooting a long gun, I shoot right handed, using my right eye. I nearly close the left eye. This does impede my peripheral vision, but lots of experimentation over the years has led me to conclude that is the only solution that works for ME.
Some people cannot close or squint one eye independently of the other. If they are cross dominant, they have to shoot long guns off the weak side shoulder.
Experiment to see how much binocularity and convergence your eyes have. It may be that you can become accustomed to using the left eye to sight with for both handguns and long guns. Everybody's eyes are different.
You can only learn from experience if you pay attention!