Wow! I admire your stamina. First of all, I would really rocommend that you get walkie talkies for the kids. This way they can find one-another and you can find them when you want too. They will work on the ship and in port and with four kids, I would think they would be mandatory. Make a hard and fast rule that each kid always has his with them...no excuses.
They will have a complete blast on the ship...guaranteed. They are the perfect ages to enjoy the freedom and activities.
For different ideas on the ports check out each port in http://www.CaribbeanPortReviews.com
In Progresso you can have fun right on the beach by the pier without spending a lot of money and your kids will love it.
In Cozumel I would recommend Mr. Sanchos, as it is kid friendly and has a great beach with no admission charge (at least the last time I was there)
In Belize I would recommend either a ship's excursion or arrange a cave tubing trip with Reggie (see Nancy N's special report on it.)
Have a great cruise and be sure to share your experiences and tips about traveling with four young men with us when you return.