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.
NCO of Marines, 3rd Award Expert Rifle, 236 KD Range
D Co, 4th CEB, Engineers UP!! OEF 21JUN-20SEP2011
REV. 19:11 And I saw heaven opened, and behold a white horse; and he that sat upon him was called Faithful and True, and in righteousness he doth judge and make war.