Neville: Yes, I do believe that MO people who may not be strict in one area will ask a shailah in the areas where they are more strict. You mentioned tznius – just remember that most of the “rules” that the yeshivish and chassidish people follow in that area are chumros, not halachos. Most MO people (at least the ones I know) follow halachah, but many don’t take on the chumros which were accepted by the yeshivish community.
AWOB: R’ Moshe was not universally followed in all areas. For example, most don’t follow his position regarding Shabbos clocks. Satmar opposed his views on many fertility treatments, and even held that many children who were the results of those treatments are considered mamzeirim. (My children, who resulted from IVF, would not be acceptable to Satmar). Many don’t follow his views regarding an Eruv in Brooklyn.