I have cruised on the Braemar several times. There are normally 2 formal nights out of 14. A lounge suit or dinner jacket acceptable for men and cocktail dress for ladies. You can always dine in the Palms cafe which serves an excellwent buffet every night and is informal or casual dress.
Try Ocean Village 1 or 2 or Island escape, or NCL Freestyle Cruises you wont need to dress up on any unless you want too.
The Braemar is a lovely ship and the staff and crew were fabulous when we cruised with them in 2002. We have been on larger ships since and always prefered the more personal touch of the Braemar. The formal nights are not every night, so you will only have to dress up a few times or if you go into certain restaurants, it will tell you in the brochure. If you do not want to dress up at all, then either NCL or Ocean Village would be your best bet, NCL are good and larger ships, mostly Americans though, which isn't a bad thing, if you like Americans, that is. Good luck and bon voyage whatever you choose.