I wouldn't say you're crazy at all. Might not be very common, but it definately does happen. How certain are you that you're right eye dominant?
I've had a somewhat similar experience. I'm a natural born lefty, but I grew up doing most everything right handed, except for writing and eating. I always shot right handed and used my right eye to sight. As I got into shooting more seriously, I eventually discovered I was left eye dominant. With that epiphany, I began experimenting shooting with my left hand. It felt much more natural and I shot a bit better too. I now shoot primarily left handed... well, handguns anyway. I still do long guns righty; just never felt comfortable shooting them on the left.
Shoot whatever way feels the best to you. Maybe practice drawing, aiming and dry-firing in front of a mirror. That's what I was doing when I began to realize my eye/hand dominance issues.
Essentially, just practice. Right, left, whatever. Take stock of what you are doing and how. It'll most likely come to you.