I'm in the same boat - basically "non-dominant" for either eye.

My eye dominance is dependent upon which side of my nose the object lies, basically whichever direction my eyes have to turn, the near side eye takes over. I've always been some blend of ambidextrous or cross-dominant with handedness (feels more natural shoot handguns left handed and rifles right, but can shoot either equally well with either hand), but my eyes ping pong back and forth... I can control it a bit by blinking or slightly closing the dominant eye - jumps right over to the other eye.

I use it to my advantage. "Co-witnessing" is inappropriately used to describe co-regulation of rifle iron sights and red-dot optics, but "co-witnessing" USED to mean the practice of letting your brain superimpose the reticle viewed in one eye to the open field viewed in your other eye.

So... What I do when I'm shooting right handed, my head naturally turns a bit to the right relative to the rifle, my eyes point a bit left, so my left eye takes over, giving me a great clarity of the open field of view. I COWITNESS the reticle in my right eye superimposed over the field view in my left to let me bring the reticle onto the target, then I blink my left eye, or slightly close it. As soon as I do that, my right eye takes over, so I'm on target with my right eye through the optic.

D@mned handy for me, in my opinion, because I can get my scope on targets in a high magnification rig way way down range without issue. For fast movers up close, I don't even need to switch to my right eye, I can shoot well enough to kill coyotes shooting with my left eye focused on the coyote outside of the scope and just letting the reticle in my right eye superimpose onto the field. It's not precision aiming, but it'll kill coyotes.