I would recommend practicing a lot. Discipline is the key in this sport. You also need to be patient as you will notice the improvement as time goes by.

I'm left eye dominant but growing up my dad had a right handed bow, so i learned to shoot right handed. There is a bit of a sight picture learning curve but I still shoot right handed to this day and I'm a pretty good shot. In fact, despite shooting a bow right handed I shoot a rifle and shotgun left handed (although I can shoot a rifle right handed as well) and I shoot a pistol right handed. I never really thought much about it and just shot the way I was most comfortable and somehow the connection between my brain and eyes compensates. I have no idea how or why...but it does. Morale of the story...it's do-able with practice.

I know this is an old thread but some of the information pertaining to it is still quite relevant. Cross eye/hand dominance is an issue with new archers. And it is easily overcome with compound bows. Shooting a bow with your weak hand is much easier to train a new shooter in. They have no muscle memory to mess the shot up. There are a couple of us in the family that have this problem. My dad was left handed but right eye dominant so he was taught shooting right handed. But back then lefties were shunned anyway so about everything he was taught was right handed. Even writing and cutting with scissors. New archers should always find their dominant eye and shoot with the hand that goes with it. Accuracy will be much improved and shooting with both eyes open will be possible.

Interestingly, I was right handed but left eye dominant. When I had my lasik done, I discussed with the Dr, that I was a hunter (was not an archer at that time) and he said he could actually correct my vision such that I would become right eye dominant.

it worked, and I've been right eye and right hand dominant ever since.