Not sure if it's considered eastern caribbean, but I would always recommend st johns. You fly into St Thomas then take the ferry over.
It is very limited in terms of shopping and touristy stuff. Though it does have a few great bars. It has all the usual island stuff, snorkeling, scuba, rental gear, charter boats. Wonderful beaches and water. I prefer staying on the eastern end of the island, as that is the quiet end.
The reason I prefer St Johns to other islands is that it is mostly national park and protected reef. To me this is why this island feels less busy than others. And since you are going in the off season you won't have to deal with as much of the Cruise boat traffic, though this is probably true of all the islands.
Depending on where you stay on the island you could do this without renting a car. A taxi to the ferry is relative cheap and the ferry is $7. The cabs on St johns will run $4-10 depending on where you want to go, they hang out at some beaches but not all, so prepare to walk if you want to explore a more far off beach. There is also a free bus that runs end to that you can catch.
The park service also provide many free or relatively inexpensive guided hikes and boat rides. With the park service I did a hike from the high point of the island down to a bay, then you take a boat back to the park building.