I'm left handed but right eye dominant. I do almost everything sports related right handed. Some I can do with both pretty well though.

I think shooting a rifle with your weak eye is much easier than shooting a bow that way. Especially if you use a scope on the gun. I always tell people unless it is nearly impossible for them to shoot a rifle right handed to go that route. It is so much easier to find rifles in right handed models. However in archery you need the right equipment if you want to be good at it.

Paul