Hi OldNSalty. Your cruising preferences sound a lot like ours. Have you considered Holland America? Their ships have a classic feel to them. While they don't market themselves as a family line, they do have a lot for families to do. Our son liked the kids' programs on our HAL cruises. And, as the ships are smaller and there are fewer families (as a % of the passengers), the kids get a great deal of attention in the programs.
Celebrity might also be a good option... yet I give HAL the edge for the kids. (In my experience, only.) Princess is superb for the family, although you'd need to get either a mini-suite or two cabins to fit four comfortably without dealing with upper berths.