I have the same "condition" as the OP.
I'm not some bigshot on the shooting circuit, but I did pretty well back in the day in USMC intramural competitions and such, particularly with the pistol. I don't think I've lost it either as I perfecto'ed the admittedly very easy Army pistol course of fire.
I learned in Boy Scout summer camp, as an 11 year old, that I am cross-eye dominant. Being right handed, I was shooting as such, and hitting very little with my single shot .22. Someone on the line suggested I shoot left handed and I shot 45/50, tying with a couple Eagle Scouts for high score that year.
I always shoot left eyed. I shoot pistols right handed and cant my head over slightly. Pistols with great triggers (single action fire in particular) I tend to score better shooting left handed due to the finer muscle memory with my left index finger.
Like a lot of lefty long gun shooters, I gravitate toward ambi guns or those great mid-20th century designs with reciprocating charging handles on the right side of the receiver. I have entirely too much muscle memory in the M16/AR family to make great use of the ambi charging handles/safeties (still trying to though). Savage, Ruger, and Tikka all make good LH bolt actions, though the Tikka is the smoothest and it's not even close.