Congratulations on your first cruis! I'm sure you will have a great time!!
If you want to keep your credit cards, cash and ship card with you thet sell waterproof cases you can wear around your neck that will keep them safe and dry. I have NEVER seen anyone from the ship come along to watch things. If you are out on a tour boat diving, snorkeling or whatever your things shold be pretty safe when you are in the water.
As for booking the excursions. I have done both and had good success. Only thing to consider, if you are late returning from an excursion you booked on your own the ship will not wait for you. Nassau is not a very big place and it would not be too difficult to get back to the ship on your own if the need arose.