As you already know, CocoCay is a private island of Royal Caribbean. So only you and the rest of the passengers can go there.
I have been to CocoCay and another private cruise ship island. When I went, I usually took as few things as possible (towel, sunscreen, hat). But most importantly, I brought my room card
(Sea Pass as it is called). Into which I placed it in a waterproof "Diving Wallet" when I went swimming. They sell these "Diving Wallets" onboard the ship (as well as your local dive shop, WalMart, etc.). Just like Sonny V. I left my belongs covered-up when I went swimming and there were no problems.
Oh, just like Sonny V., I also brought along my camera and snapped this shot:
So, have a good time on your cruise