We went on our own to Orient Beach in St. Martin, just picked up a taxi downtown after shopping. The occasional naked man or topless woman walked by on this beach, so if that's something you want to avoid, I'm told the beaches on the Dutch side have less/no nudity. For two people, I figure I saved about $40 by doing the excursion on my own. Plenty of taxis are available to take you wherever you want to go.
You might try giving Disney a call to see what excursions are available. I believe that for some excursions - and beaches might be that type - they take almost an unlimited number of people, since they can always add another bus or van. The only Disney excursion we did was the Captain Nautilus speedboat snorkeling at St. Thomas/St. John, which we loved. Like you, I procrastinated and didn't call until the Monday before the cruise. It wasn't available, but I called back on Thursday morning, the last day for people to cancel excursions without penalty, and we were able to book because they'd had cancellations.