I would repeat the north part of the Exumas and then go to Cat Island and Eleuthera which are never mobbed like the Exumas or Abacos. There are morning flights out of both of those to Nassau that connect to US flights. No need to make reservations until a couple of days before the flight when you'll know where you'll be.
Skip Cat Island if there's a chance of a west
wind as there aren't safe harbors but in a prevailing easterly just pull up to the beach and drop
anchor. New Bight is especially fun with a grocery store and several restaurants. You can climb the "hill" to the famous monastery ruin. Hidden Treasures is my favorite restaurant. Tables right on the sand. You'll never forget the fresh made Mango Daiquiris. Cat Island farmers grow huge quantities of tomatoes and fruit for export so you eat well.
Eleuthera has three airports with flights. In the south an excellent marina resort at Cape Euethra Resort or go around the point to Rock Sound with many restaurants and grocery store. Beautiful Governor's Harbor is your next stop with some great restaurants and
shopping. In the north, Spanish Wells is worth several days either in the excellent marina or a
mooring.