Also, I'm left-handed but right-eye dominant. Am I correct in thinking that I should be shooting right-handed?
Not necessarily. Of course, it would be wise to find a weapon that is considered "ambidextrous" in nature, in terms of mag release, safety, etc.
Now, back to the cross dominance issue. I myself am right handed but left eye dominant. While I do practice shooting left handed, I still mainly shoot right handed...I simply turn my head to my right shoulder, thus aligning my left eye with the sights. It takes a bit of practice, but is a relatively easy habit to acquire. I would imagine things would be the same in your case; just mirrored, of course. From what I gather, this technique is easiest with an "isosceles" stance. But, I have also experienced great success using the "modified Weaver" stance (or Chapman, to be specific). This is my favorite.
Also, you could simply close your non-dominant eye, but I generally think it is best to train with both eyes open...particularly for HD. Now, a half-squint (of the non-dominant eye) may help, all while still somewhat maintaining your peripheral vision, but I generally think this technique is best saved for a long gun (I have used this technique with success, for my shotgun, over a year now). So, to reiterate, unless you do not mind the struggle with switching your strong side, I would suggest starting out with a simple tilting of the head.
But, again, given that you are left handed, pay attention to the layout of your next gun. This can either make things easier or a bit more difficult. Frankly, this is one reason why I like the XD platform....ambidextrous mag release and no reliance on a thumb safety (unless you specifically buy one with such a configuration).