I guess I'm on the casual side of the fence with the dress suggestions. I haven't dressed formally in over ten years & have never had much problems with the results. I always wear a long sleeve dress shirt with dress pants without any jacket or tie in the main dining room and they never refuse to seat me. It's very comfortable, easy packing & takes nothing away form anyone who wishes to dress as formally as they want.
When they begin to serve the same dishes at the buffet on formal nights as in the dining room I'll stop being attracted to eat there but as it stands now all the items I like most are served on formal nights in the DR.
Although st people still dress in the DR's, Princess has defitnely become more casual over the years as the luggage costs have gone up. Dress nicely & you'll have no problems.
btw- The Specialty restaurants are always casual even on formal evenings.