Hello Fuzzy,
I sympathize with you. I'm going through the same thing now. It sure explains why I always had to close one eye to hit something! Since I switched sides my aiming is lots better just by being able to keep both eyes open. I started with throwing lefty to start training that side more and I also have a little ball I've been tossing up and catching just training the hand eye coordination on the left side. I got a left handed bow but I overbowed myself...now I'm looking for a lighter left handed bow to work out the form on before I move back to the heavier one. Just like with your firearms marksmanship, it just takes patience and practice to get the brain and body used to the switch. Just don't get too much bow to start with. Go down to the archery store and try a few to get an idea of the comfortable weight to start with. You'll want to figure draw length and as well as how long of a bow is comfortable to you also. The shop should be able to help you with this.
Good luck. Lots of good info here.