Your DH does not need a jacket for meals in the dining room. Formal Dress up nights are Sept.19th and Sept.26th. Dress on these nights vary with the individual. Some men will wear a dress shirt with or without a tie and or jacket. Some will wear a suit. Some will wear a tux. It is an opportunity to "dress" for dinner. We have seen a bit of everything. My DH will be wearing a tux. A suit is fine.
Other nights are cruise casual which is usually a polo shirt or shirt wth a collar and a pair of dockers. Dress Jeans are permitted in the dining room.
Tropical night is Sept.20th. Colorful, tropical casual, comfortable, happy attire is the norm.
Pirate night is Sept.23rd. Just about anything goes here Matey! Costumes, T-shirts, pirate gear of all makes and models. cruise casual to cruise crazy neon. Your server will provide you with a pirate bandana and even help you tie it. No worries.
As for being underdressed unless he arrives in his PJ's other than on the 19th for breakfast or for dinner in tank tops with torn shorts and flip flops I'm sure he will be just fine