Nate,
Usually, with a rifle or shotgun, right-handed/left-eyed people can achieve better marksmanship by simply shooting left-handed. Obviously, the opposite is true for left-handed/right-eyed people... shooting right-handed is usually best.
With a handgun, this may or may not be true - usually, the natural instinct will be to align the sights using your dominant eye, and it is easy enough to slightly change the position of your head to align the dominant eye with the sights. Try both ways and figure out what works best.
Just curious - how do you shoot if/when you play billiards? I took a billiards class in college, and my partner was right-handed/left-eyed. Pool was very, very difficult for him because of this.