re: Educate me on Exuma, Bahamas

We did the opposite. Rented a house for nearly 2 weeks and spent 1 of those days at Sandals, just for the day. Spent the remainder of the time exploring Exuma and small islands surrounding it.
I would recommend allotting more time for DYI time away from the resort, but that's just my opinion. A few in our group didn't do much exploring, they wanted to simply relax, but had a good time when they did get out. The people of Exuma are extremely friendly and the island has a low crime rate. The roads they do have a terrible (by our standards), the groceries are really expensive and hard to get at times, as the shipments only come twice a week to stock stores.

Try a tour to pig beach using one of the smaller outfits, the time spent with the pigs is only an hour or so, but they take you to a number of other attractions only accessible by boat. The smaller the boat, smaller the group, and the more they can customize the stops to do what your group wants.

There are lots of beaches with road access along Queens Hwy, and nearly all beaches, by law, are public, just need access. The out islands have some of the best beaches in all of the Caribbean, so spending your time there and only seeing one of them would be doing yourself a disservice.