I am extremely overweight - and enjoy cruising a lot. Every ship I have ever been on has armless chairs in the dining rooms. We ask the Maitre' d before our first meal and they have always taken care of it.
Morbid obesity is considered a handicap according to the ADA and you can feel free to request a handicapped accessible room.
On RCCL voyager class ships this is a must as the shower doors on a regular room are too small for most very overweight passengers. They have roll in showers with large fold down seats.
Johnny Rockets is booth and stools only inside on voyager, but you can sit outside at a table.
I sit in my wheelchair for most shows and I would say that the theater seats could be a problem on some ships.
I am leaving in 7 days for a RCCL voyager class trip and plan to have a great time with a few small concessions on my part.