Hi John,
As John mentions, wandering around the old Navy station is pretty interesting, and that beach over there is beautiful.. I've never seen another person on that beach...
If you have the time when you leave Alabaster Bay, and conditions are favorable, you might make a stop in the little settlement of James Cistern... I get the impression very few cruisers stop there, but I found it to be a very friendly, welcoming place - nice place to wander around, maybe have a breakfast or lunch... The town is renowned for their girl's softball team, they have a really winning tradition there, it would be wonderful to see a game there in the nice little ballpark...
Rainbow Cay is another nice spot if conditions are settled, and you're looking for an alternative to Hatchet Bay... Really nice little restaurant nearby, the Rainbow Cay Inn if memory serves? Been several years since I was there, however, of course it could have all changed in the interim...
best regards,
Jon