I'm cross eye dominant also. It's never caused me an second of concern shooting.
With a pistol I shoot Quell Stance both eyes open.
With a rifle with irons or a magnifying optic I shoot with my left eye closed and right picking up the sight picture.
With a red dot/holo sight I shoot with both eyes open.
In all cases I use my dominate hand on the trigger except for occasional barricade shooting etc.
You can also check out the Bindon Aiming Concept. That's both eyes open with a magnifying optic.
Airsoft, where nothing is hurt but feelings.