I am also very right-handed and very left-eyed. I am a photographer by trade and have always instinctively lifted the camera to my left eye. When I took up competitive shooting at the tender age of 53, I started righteyed as coached. At a certain point I realized I had better sight picture with my left eye. With the help of a coach I adopted a modified Weaver stance with my feet and trunk canted to the right (when shooting freestyle). During one-handed strings I do the same cant with my free hand balled to my chest. Ironically. This results in better scores during weak-hand strings than strong-hand ones. I'll echo those above, I always keep both eyes open but turn my head enough that my right eye is away from the target. A friend who I often squad with puts a post it note over his non-domination eye!