Definitely take cash into Nassau if you want to shop. They will tell you not to take very much. They also tell you that you can use travelers checks and credit cards. It seemed everywhere we wanted to get something they would only take cash. We even stopped in at the tourist info booth in the building you walk through to shopping. They would not even cash a travelers check for us. We ended up going back onto the ship for more cash. Luckily, the ship is right there. It took us about 15 min. to go back, cash a travelers check at guest services and go back. It was extremely hot and we almost didn't go back.
Have fun!