Bear in mind that I am new but I too am cross eye dominant.
Left eye dom, righty. I tried shooting on my right side with my left eye closed, left eye taped, both eyes open, etc. After shooting for about a month (with both eyes open) I was shooting in the high teens and twice hit 21.
I was happy w/ my scores but with both eyes open my left eye was taking over and when I missed I was never sure why. My sight picture was the target well to the left of my barrel. My barrel was aligned with my right eye. I knew I needed to change before too much time had elapsed.
I took a lesson with Bruce Ney and he convinced me I'll be better off in the long run switching to my left side. A few pointing exercises also made the point very clearly. I said "ok, I'm convinced" and had him bend my stock to better fit my impending left sided mount. There was no turning back now.....
So, I got the gun back Wednesday, took a lesson, worked on "looking through the barrel", my sight picture, and my mount.
My first day out was today and I shot 17 and 18. I'm thrilled. The change wasn't as bad as I thought, I hit about the same number as I was from my right side and when I missed I knew why.
I'm happy I switched to lefty and won't be switching back. I'm sure my scores will continue to improve.
Just my experience as one who also struggled with what to do about cross eye dominance.
Hope it helps.