I second boat Boatpoaker's recommendation for Manjack. Many long beaches, often with nobody on them and only a few miles from New Plymouth on Green Turtle.
A4 is a popular anchorage. I've often had A1 to myself. (It's not where you want to be if a cold front come through!) A2 is fairly good in all weather conditions if you have the draft
to get into the bay a ways, and has easy access to the trails to all beaches.
No Name Cay just south of Green Turtle is also uninhabited, but the anchorage is more exposed, and gets more boat traffic, including day trippers that stop by to feed the family of pigs. (That recently had piglets) They like apples if you stop by.
Anchored at both Manjack and No Name just last week.
Have crossed the gulf stream
several times in 26-footers. I agree that the conditions matter more than the boat.
Other places in the Abacos I often have a beach or anchorage to myself:
Joe's Cay and Jack Cay in the Carter's Group.
- Ocean side bay.
Snake/Deep Sea Cays