Hi Try Independence from Southampton it will be full of kids but there is plenty to do as each age group has its own club first class shows including an ice show and plenty for adults as well but book as early as possible you may get a free child place if you shop around.
Whilst it is generally accepted that RCI offer the best cruises for families, unfortunately for you Independence otS is sold out of all 3/4 berth rooms for the whole summer period, and has been for several months. In the Med Brilliance otS and Legend otS are also sold out for the summer season - your best bet is to look at 7n cruises on Navigator otS or Voyager otS. You will still get the big ship wow factor with them.
Try the Voyager from Barcelona,cheap flights, easy transfer to port, great ship, lots of facilities excellent entertainment.