I'm cross-dominant, also. I grew up shooting right handed, and never had a problem with rifles and pistols. Apparently, that's because I learned to close my left eye while sighting.
Then I took up the shotgun sports, and had a real problem. You can NOT be successful at trap or skeet unless you keep both eyes open. With a dominant left eye and the gun on the right, however, it is almost impossible to shoot with both eyes open.
There are only two solutions. The best solution is, as others have pointed out, learning to shoot from the side of your dominant eye. It isn't hard, but it does take practice -- LOTS of practice. You need to practice daily for weeks, if not months. It doesn't have to be live fire every day, though. Dry fire practice helps, too. I had a shotgun coach who advised me to keep my shotgun in the living room, and every time I walked by it I was to stop and practice some mounts and swings on the left side. By doing that every day, dozens of times a day, it became 2nd nature fairly quickly.
The other solution is to trick your mind into switching dominance to the right eye. You can do that by closing the left eye, obviously, but that has the disadvantage of eliminating a lot of peripheral vision and destroying depth perception. That's why you can't do that in shotgunning.
The old shotgunners trick is to smear a tiny amount of vaseline on the left lens of your shooting glasses, right in the line of the left eyes sight when the shotgun is mounted and you are looking down the barrel. The small amount of vaseline in that spot blurs the left eye's picture just enough that the brain switches dominance temporarily to the right side. That way, a left-eye-dominant individual can shoot from the right side with both eyes open. Of course, it only works when you are wearing those glasses, and looking too far in any direction other than straight ahead for too long will cause dominance to switch back to the left. Thus, it's a stop-gap measure, and not a permanent fix.