We're on the Sea the week before you are -- May 14 - 21, and we also cruised on the Sea in January, 2004. We have done 2 Mayan ruin excursions through NCL --- yes, they are pricey. In Cozumel, we went to Tulum. I'm sure it's possible to do it as an independent tour, but you do have to ride the ferry from Cozumel to Playa del Carmen. When we went, it was our first cruise, and I wasn't comfortable booking independently -- especially for a fairly long, involved excursion.
We also went to Altun Ha in Belize -- it actually wasn't that much money. We hadn't originally planned on getting off the ship in Belize, but decided at the last minute to do that one. Once again, as first time cruisers, we just felt safer going through the ship. I'm sure there are private excursions available here. You do tender into Belize, but the actual excursion time is short enough that I might take a chance doing that one independently.
This year, we're planning on doing the Lamanai/River Safari trip. That one is 7 1/2 hours, and there is no way I'd consider doing it except as a ship excursion. There just isn't enough extra room in port to make me comfortable going with someone else.
When we went to Belize, there were a number of people offering side trips to different places right off the tender dock. My wife and I took a side trip with one of the people there offering a trip to the ruins for about half of what the ship charged. He had a small bus, and he offered to take us if he could find 8 people to go. He left for just about 10 minutes, and came back with another six people in tow, and off we went. We made sure that we would get back to the ship in plenty of time, and we did. When we got back to the ship an compared notes, it seemed that we had one of the better tours.