We did Royal Palms as well a few weeks ago. We decided to go there since other folks on the bus were going there as well.
Waiting for a taxi was taking too long so I just hopped onto van (which happened to be a bus) that had enough room for me, my wife, DD5, and DS3. It was $2.50 a person.
We arrived at Royal Palms at around 12:30. The kids had no admission fee. We were able to get 1 lounger on the beach, which was really a place to store all of our stuff while we were in the water.
They have changing rooms/bathrooms at the facility.
On the way back to port they have waiting taxis to load you in, and that was $4 a person.
When we head out on our next cruise next year we plan on doing the same thing. The only difference is we will try to get off the boat earlier, and not have reservations at 3 on the boat so we can enjoy the beach longer.
Cheers!