R.C.C.L. or Princess are probably your best bets. We went on the Splendour of the Seas (Royal Carribean) out of Vanice and had a wonderful time. There were of course the obligatory "formal nights" but generally the cruise was quite informal and a hell of a lot of fun.
Don't forget it is very possible to eat informally even on "formal nights" on most all cruise lines. Generally there is an informal dining room or a dinner buffet available. You can also have a romantic dinner brought into your room by room service, perfect for a pajama party or just a quiet dinner.
The choices are all your own.