We were just talking about this when I went through my CCW class this past Saturday. This is where I discovered that I was cross-eye dominant (left eye- right hand) as well.
I don't have any words of wisdom when it comes to long guns, but what I have found with shooting a pistol that works for me, is to use an exaggerated Weaver stance to the point where I'm almost standing side on to my target, which in turn brings the weapon up to the proper sight picture.
Once I realized why I kept pulling most of my rounds to the left (thought it had something to do with grip, anticipation, etc) my accuracy improved significantly. It's still a work in progress, but I am getting better.
Knowing that there is an issue is the first step to either over coming or correcting it.
good luck to the OP
Exco