I really can't believe some of the things I've read in this thread:
If you're left-eye dominant and right-handed, you should switch to shooting with your left hand? WTF?
People aim with only one eye? Really? Do those people drive with one eye, or watch TV with one eye? Do pitchers and 3-point shooters aim with one eye?
You can't have eye dominance and play with the chin under the middle of your cue? Again, WTF?
Johndagolfer, if your target is becoming a double-image, you are not focusing properly. Your eyes are either focusing on a point forward or behind your intended aim point. This causes a parallax view, which is your double image. Try not to stare at your target too long - the human eye can only focus for so long (no more than 2 seconds for most people) at a single target before it wants to wander. You also must give your eyes enough time to acquire the target (around 1 second for most people).
As tedkaufman said, cross-dominance is not a big deal - there are plenty of players like that. You DO NOT have to line up with the cue under your dominant eye. You also DO NOT have to line up with the cue under the middle of your chin. Comfort, and what helps you see the intended line, are the important factors. EYE DOMINANCE DOES NOT MATTER. You can line up with the cue under your ear (see Neils Feijen and Earl Strickland), under the outside of your dominant eye (see Karen Corr), under the inside of your dominant eye (see Nick Varner), under the middle of your chin, under your non-dominant eye, behind your back, wherever. Finding what helps you see the intended line is the hard part, and like Flex said, Joe T's device can help immensely with that.
I have one piece of advice for the OP - get thee to a certified instructor immediately. The garbage in this thread will only make your game worse. Sifting through all the crap just isn't worth it.
-djb