Laura,
Pull-ups are done with your palms facing away from you on the bar, and chin-ups are done with your palms facing you. You get to use a little bicep with chin-ups. In P90X, he tells and shows which ones to do.
My body sways forward a bit as well, and I have a real tall bar. Don't know if this is real normal, but watching the folks on P90X, I think it is ok. They all seem to do it, too.
Julie