I have the same left eye / right hand dominance. Many years ago I started shooting from my left shoulder. Its not nearly as difficult as one might think and soon it is the norm and trying too shoot right handed feels weird. For me it was the best long term solution and possibly is something to consider. Cheers!
Although left eye/right hander, as noted; I shoot right on rifles, and use my right eye.
Handguns, I use my left eye when shooting.
Just something I acquired over time.
I am the same as above. I started shooting everything right-handed, as I am right handed, and I am also left-eye dominant.
When shooting a rifle, I shoot it right handed. With a scope infront of me, I have no problems focusing through the scope with my right eye. This is a bit harder with iron sights, but still very much doable for me.
With a handgun, I hold it right handed, but usually (not always, Im still a total newb with a handgun) raise it in line with my left eye. Makes for an easy enough fix.
With a shotgun, I just shoot it left handed. Started with a pump, now I have a cheap Semi. It ejects out the right side, but it doesn't come close to hitting me.
I tried a bunch of tricks, including closing one eye, using scotch tape over my glasses on the left side, using sunglasses over just my left eye, and trying to focus with my right eye. Long story short, nothing worked as good as I wanted. You need to have both eyes open for two reasons: First, you need to see the target as soon as possible, second, having both eyes open gives your brain WAY more info about the speed and trajectory of an object (Especially for the ones coming towards/away, but crossers too. You have no depth perception with only one eye open)
The FIRST day I shot left handed, I knew it was the right choice. The gun felt awkward, heavy, and too long when I first switched. I dont know why, but it seems my comfortable length of pull on my left side is significantly less than my right (like 3/4" or so) and I felt like the pump was a mile away shooting left handed. (Part of the reason I picked up a semi)
I only switched in the early parts of this year, and I haven't spent a lot of time shotgunning this year (didn't even get a range membership due to money and time) but in maybe 200 clays (3 rounds of skeet with my mossy pump, and probably 125ish hand thrown or launched with a cheap $40 trap machine) I already feel way more confident in my abilities.
Shoot your shotgun on whatever hand is your dominant eye. You can re-learn to shoot a shotgun with your left hand, its much harder (personally I think impossible) to reteach your eyes. All the tricks out there might increase your success rate, but I don't believe you'll ever hit your true potential by shooting on the side of your non-dominant eye.