Robbor,
I'm left eye dominant and spent 45 years shooting right handed and closing my left eye. I count number of boxes of shells per duck felled. I started shooting F class at 300 meters with a local club about 3 years ago. Most of the shooters are 3 position and a few have won medals at nationals, internationals and olympic matches. I tried taping over my left lens, tried a shield on the sights. All that happened was I still did not have a field of view worth anything, couldn't simultaneously see the target and wind flags, and still had headaches. One of the better shooters suggested I shoot left handed. Ha. How do I do that with a right handed rifle? Plus, I'm too old and ornery to change now.
He finally convinced me to try it. F class is prone which is pretty simple to try the switch, no sling, no kneeling, no standing, etc. I moved over to the other side of the rifle. He helped me with my body position. It felt weird.....for about 5 minutes.
I discovered a whole different world out there. I could keep both eyes opened, see if I was shooting on the correct target,usually a good thing), see wind flags I knew were out there, no headaches, plus my left eye is better, so even the target alignment was better. The only strange thing is that I have to work the bolt with my left hand by going over the action to operate it; my right hand stays out front so as not to "break" my position.
We don't have a lot of F class shooters but since switching to left hand shooting, I've never been beaten.
Oh, I recently purchased my first left hand rifle, an Anschutz .22 and have started to work on 3 position shooting. I can now shoot poorly,not awful) in kneeling and standing with the .22.
My strong advice, switch to left handed shooting and a whole new world will appear.