I am the same way. You are right handed but left eye dominant, keep shooting left handed, do not fret. This can actually be an advantage. I personally have never bothered with left handed rifles, I don't like them. Ive always used right handed rifles and gotten used to it.
With a bolt action rifle firing from a supported position such as a bipod it means you can keep your firing hand in place, your eyes somewhat in the scope, and work the action all at the same time.
With an M16/AR15 it means you are actually using you strong hand to perform a tactical reload. I was much faster than most guys in my platoon and even many infantry guys just due to this simple fact.
I'm sure there are others that I don't know about.
Sgt. of Marines, 5th Award Expert Rifle, 237/250
Expert Pistol, 382/400. D Co, 4th CEB, Engineers UP!!
If you start a thread, be active in it. Don't leave us hanging.
OEF 2011 Sangin, Afg. Molon Labe