Well we've not done Disney, but have done Carnival. I have had luck with NCL and RCCL too. both have fine kid's programs, especially for the 7 year old. I think the 11 year old would be happy too.
If you choose NCL, budget for a larger room with a balcony (I like the minisuites and they are usually not much more than a regular balcony) also budget so you can try the surcharge restaurants. Make sure you book one of their newer ships The Pearl, Gem or Jewel and I think you'll be happy. The Spirit is a fun older ship, but they do not have any quads with balconys, and the inside and porthole quads are microscopically small.
Royal Caribbean is a fine cruise line as well. Their larger newer ships are beautiful from the outside and look like they have lots to offer. I have been on ships docked next to them. the Enchantment is a nice older ship and I wouldn't mind sailing her again if she would do a few trips longer than 5 days.
I really do not like to fight the crowds either, and understand.