Hello Tsatt,
You said: "we most likely will not get to eat in the dining room." From that it sounds like you have the wrong impression of what anytime dining is. Anytime dining only refers to you not having an assigned time and table to eat. It does not mean eating in the lido deck buffet.
There will be three main dining rooms on the ship, all very much alike. All having the same menu. The traditional diners will use just one of the dining rooms. The other two are for the anytime diners. So you will get to eat in the dining room.
All that's different is that you go to one of those two anytime dining rooms whenever you are ready to eat. You can choose to sit with other passengers, or ask for a table just for you two. You can eat early or late as you like.
The draw backs to having anytime dining is that you may have to wait to be seated, just as you would in a regular land based restaurant. That's more likely to occur at the busy dinner times. Earlier or later, and you will likely be seated immediately.
If you know ahead of time, when you wish to eat you can make a reservation in one of those restaurants, and your table will be ready for you.
Some of us really enjoy sitting with other cruisers for meals. The anytime diners do miss getting to know their tablemates the way you would, were you eating with them all week. But if you choose to sit with other passengers each night, you will get to meet a lot more people, which is fun too.
So go ahead and pack your formal duds. You will be eating in the dining room.
BTW, There is likely to be a few other specialty ($) restaurants on most Princess ships, that are even dressier than the main restaurants, and you may want to splurge on that experience once during the cruise too.
