I have been to Nassau 5 times and can say: I have NEVER been to Paradise Island, Atlantis, the Straw Market, dolphin stuff or any of the party boat excursions. I also did not stay on the ship. This past trip I went snorkeling with Stuart Cove's on the south side of the island, and it was great! I've stayed at Sandals Royal Bahamian, and I am pretty sure you can get a day pass there (but they don't allow kids). A really cool place to visit is Compass Point on Love Beach for a funky island vibe: drinks, lunch and people watching.
Just get out of Nassau and visit New Providence Island. (Nassau is the name of the city, not the island.) There are lots of nice beaches, hotels with day passes, funky restaurants, small charter sailboats, etc.
And if you are a diver, Stuart Cove's operation is great. I've dived with wild dolphins and sharks there! And turtles! That beats shopping with thousands of tourists any day!
Have fun whatever you choose to do! Staying on board and chilling out is great too!