I would suggest visiting Seagrape Beach at the far end of 7-Mile Beach. There is a restaurant, bar, lounge chairs and plenty of beach in both directions if you want to get off by yourself. Most of the hotels will be full then, so they are bound to be crowded. You can get a van to take you to SeaGrape for about $5.00 per person from the transfer stop to the left of the tender terminal. Everyone that I know that has spent the day at Seagrape Beach loves it.
Hope that helps.