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Sounds like Ventura might be a good bet. Most big ships cater for all ages. Beware of small ships which do not have the same range of activities and can result in bored kids.
You could also consider Independence of the Sea. There is so much for children on that ship and they also have family cabins which could suit you. Keep your eye for Oasis too. ... Neil.
RCI are generally acknowledged to be the best ships afloat for children. You could try a family cabin that sleeps 6 but that does only give you one bathroom (!). Otherwise try for adjoining cabins. However Independence is totally sold out for rooms sleeping more than 2 in the school holidays, when I assume you want to travel. So perhaps look at Navigator or Voyager out of Barcelona or Rome. Slightly smaller but good facilities and you may still be able to get on for the school holidays.