No one wants to go looking for a different bow, and you need to try to find a cheap, lightweight one. Right now, before you even get started looking for a bow, begin pulling your bow with the other hand—right now. Work at getting it to full draw. I wouldn’t even put an arrow on the string. Think about who you know that shoots left-handed, and ask them about something lightweight. Do lots of shooting up very close—nothing over 10-15 yards—until it begins to feel right (correct). Good luck, and keep me posted on your progress.
Don Thomas: Cross-eye dominance can be a frustrating problem, and it’s more common than most people realize. In the wing-shooting world, with which I am quite familiar, the conventional advice is to follow the eye and not the hand—in other words, a right-handed, left eye dominant shooter should simply learn to shoot left handed. That’s easier with a shotgun than a bow, but it can still be done and I know a number of people who have done so successfully.
I would advise against closing one eye if you shoot instinctively (no reference point on the arrow), because that will cost you your depth perception and require you to master some sort of point of aim shooting method.
The young son of a friend who grew up bowhunting with his father and me had this problem (left eye, right hand dominant), which he solved by shooting right-handed but turning his head at full draw so that his anchor was below his left eye rather than his right. He has grown up to become an excellent shot and highly successful bowhunter.
Good luck with whatever solution works best for you.