Both the wife and I are cross-dominant (her RH,LE & me LH,RE).
First me:
Shooting rifle for me is no issue at all, since as kids we ALL held our Daisy air rifles right-handed and it just feels natural. Pistols are shot lefty with a slight tilt to the head to get proper sight alignment with the right eye. I tried both eyes open, but occluding the non-dominant eye, or closing it works best for me. Also some windage adjustment of the rear sight to achieve proper alignment has helped. As I shoot competitive Bullseye, I've needed to find how best to deal with this and for that I've always used a dot sight, where there is no issue with use of either eye. For the two-handed, iron sight stuff, it's head tilt, adjusted sights, and occlude or close the eye.
Now her:
She's a very casual, occasional shooter, but success didn't come until she adjusted to the cross-dominance issue. Shooting with a dot was immediately successful, but irons led to some experimentation, which included switching to LH hold (some success, but not comfortable), occluding/closing the dominant eye (some success, but again not comfortable), and the tilted head, closed eye, adjusted sight method (most success and comfort, and, IMO, the best real-world solution if she had to use a firearm under stress for protection.
This is why my latest acquisition was a SW637 with CT laser grips. Best solution for the occasional shooter in a self defense situation (again IMO). She's never tried rifle, so we'll see about that another time.