Due to newer optics the last couple of years I've been able to shoot with both eyes open on some of my long guns. I have an Eotech and a red dot on 2 rifles which lets me shoot with both eyes open. I still close one eye with handgun and standard scopes. So I guess I'd suggest looking into some different optics to help you. I'm sure someone with more expertise will be along. I have the same right handed-left eye dominant issue.