There are no formal nights on Azamara, on NCL it is Freestlye so you don't have to dress up if you don't want to. I think that the 3* brands like Ocean Village don't have formal nights eithe.
Ocean Village is completely informal. No dressing up required at all. You can dress up a little to go to James Martin's onboard restaurant but over dress and you'd feel out of plac.
Oceania Cruises is casual. Also Regent Seven Seas Cruises have no formal nights on their 7 night cruises in the Med or Alaska. No cruise line can enforce what you wear just as long as you don't turn up in jeans or shorts at a formal do. If you don't want to dress up, go to the buffet on the formal night, not the restaurants.
Thomson cruises only have one formal night per 7 days when most people dress up, but some choose not to and eat in the informal dining areas, they only insist that you do not wear shorts, costumes etc in restaurants on evening and ask you to be covered up at buffet stations for obvious reasons.