I'm right handed left eye dominant. For rifles/shotguns I just close or squint my left eye. It works good enough for my target shooting and hunting.

For handguns I tilt or turn my head to the right and move my arms to look with me left eye while shooting right handed. I tried shooting left-handed, it just doesn't work for me. I don't do competitive handgun shooting though.

I am exceptionally lucky on this topic. I grew up mostly shooting shotguns, and never knew I should be able to keep both eyes open. Strongly right handed, kept left eye closed. In my twenties, I started shooting pistols, and had to either keep my L eye closed, or push the pistol to the L to line the sights up with my L eye with both eyes open. Started shooting some rifles as well, and reading this forum and learned about cross dominance. Figured I’d just have to keep my L eye closed shooting.
Fast forward to about 2013 or so, and I had LASIK eye surgery, and suddenly my L eye was 20/20, but my R eye corrected to 20/15. And immediately, I was R eye dominant. I can now shoot with both eyes open, but my R eye and R hand work together (and my shotgun shooting has significantly improved with this newfound ability to shoot both eyes open!)
Every since getting glasses at 10yo, I felt my L eye was better than my R. I suspect I was always R eye dominant, but just had better vision with my L eye, and hence it kind of took over. Once that switched, the R eye was back in play.