The real answer I’ve seen work well (and with almost immediate result) to help train truly cross-eye/hand dominant people is to teach them to shoot with their head slightly tilted to advantage their dominant eye. A right-handed but left-eyed shooter, even though shooting both eyes open, should tilt their head slightly to the right (I’m not talking about an exaggerated crank or lean, see photo) which gives the dominant left eye better view and focus on the front sight. The right eye is still in play to assist with viewing of the sights. Reverse this instruction for left handed but right-eyed shooters.
Turning the head upright results in the weak eye now being off the sights and looking off target. Bad idea. Tilt, don’t turn.
Then go spend thousands of rounds of ammo training yourself to tilt your cross dominant eye into play.
Sorry, there’s no free lunch.