I haven't been to Chankanaab (?) in Cozumel, but I have been to both Paradise Beach and Mr. Sanchos' Beach Club. For Paradise Beach, there is no entry fee, as the impression is you'll spend money while you're there. Lounge chair use is $2 US. And for $12, you can use their snorkel equipment, beach mats, and aqua park. This is just a bunch of water inflatables that they put in the water. There's other activities to do there as well. I did parasailing, which was great. If you do that (regardless of where you go), you should be able to see the ships docked a few miles down, if they take you up high enough.
I did this similar itinerary back in October of 2010, only we docked in Ocho Rios, as opposed to Falmouth. Here are my opinions.
Labadee - as you stated, this is a good port for a beach day. I did the zip line, parasailing, and the roller coaster, which is a bit of a walk from where the ship docks, but again, with the age of your kids, I'm not sure if they'd be old enough. Lunch this day will be on the island, as this is Royal Caribbean's private island. It's a barbecue style lunch, with hamburgers, hot dogs, corn on the cob, etc.
Falmouth - Definitely consider a ship's tour at this stop, as they have contracts with reputable tour operators. If those tours look expensive, then you may want to try shore trips.com, as they are another operator who has looked at the tour operators and went with ones that are reputable and safe.
Grand Cayman - My only trip here, I did a Stingray City excursion. This is a trip out to an area where there are lots of Stingrays, and you can touch them, feed them, etc. If you don't want to do a ship's excursion there, then just walk off the tender and see what options there are once you get off the boat. Grand Cayman is a tender port, where you have to go down to deck 1 on the ship, and take small boats, or tenders, to shore, as the water is too shallow close to shore for the big cruise ships. There is a chance, however, that you may not be able to get into Grand Cayman if the weather is really rough. This happened to me on the last trip. It was raining, windy, and choppy seas, and the port of Grand Cayman deemed it too dangerous for the tenders to run. If this happens, check with the companies you booked excursions with to see what their refund policy is. I booked an excursion through the ship, which was refunded by means of an onboard credit.
Cozumel - Another good port for a Beach Day, and like I said, Paradise Beach would probably be my preference, but do your research, and see what appeals to you. Cabs are plentiful in Cozumel, to get from the pier to the beaches.
The ship - Freedom is a great ship, and is one of the larger of Royal Caribbean's ships. You can read my review of the ship here