Whether you take your own towels or use
DCL's, you will still have to transport them to your excursion from the ship, so I would plan on a beach bag big enough to carry towels.
We were given towels for our DCL excursion while waiting in line to get off the ship. They just brought down a big rack of pool towels and let you take what you wanted. If you do an excursion on your own (like we have done too), then you can just go to the pool area and pick up some towels to take with you - just make sure you bring them back. There has always been a big bin to put the used towels in as you return from your excursions. On
Castaway Cay, there are towels as soon as you get off the ship and a bin when you get back on (aong with
hot washcloths).
As for passports, there has been many discussions on this (especially since passports are not required for the cruise). The purpose of having a passport is to make it easier if you have to leave a foreign port on a plane in an emergency. That said, if your passports are on the ship when you need them (like locked in the safe), then you might not be able to get them off if there is a true emergency late in the day. If you take them with you, there is always a chance of losing them or getting them stolen. It is your decision.
As for other stuff, since we don't know where you are going, it is kinda hard to suggest things. I would always take sunscreen, a small first aid kit (a couple of band-aids and neosporin), sunglasses, money (I keep in several places just in case), ship's contact information from that port (on the navigator usually), insurance information (since I always get private insurance), swim goggles, a small bottle of contact lens solution, and a small sealed snack (like a trailmix bar - if allowed). For the insurance information, I always make a small card before I leave home and laminate it to keep with me.
Hope this helps.