I know you got plenty of answers already, but let me just emphasize - yes, you'll have plenty of time to do things together as a family - up to 100% if you want! And, if and when you do decide to split up, you know the kids will be very close - the thing that took getting used to for me on the cruise ships was that you were never more than a few minutes walk from any other place on the ship.
We did the Alaska trip last summer. For me, my wife and I almost never have time for just the two of us, so being able to have dinner and brunch in Palo, plus a couple of other times when the kids were in the clubs, was really valuable. But, we still spent the vast majority of time together as a family - basically all the meals (except the two Palo meals), the shows, the shore excursions, and other activities on board from movies to just watching the scenery. Kids did visit the clubs almost every day, but the trip still felt mainly like a family trip.
The biggest problem will probably be convincing your kids to spend time with you and not in the clubs. You might want to make sure you set expectations for them ahead of time.