Yes, you have to take your towels with you. We were on a Royal Caribbean cruise and just took the towels the ship gave us. We just made sure we brought them back because I think you have to pay $20 for each unreturned towel.
I'm not sure about DCL's policy re towels at port. I thought I had read somewhere that they did provide them but I'm not 100% sure. Hopefully someone else can let you know about this.
You'll love Paradise Beach! It's a wonderful place! Have fun!
If you book a DCL excursion they will give them to you as you leave for your excursion, if you are going on your own, they are handing them out right as you get off the ship. But yes, DCL provides you with towels. Not huge ones to lay on, but towels.
GET UP TO A $1000 SHIPBOARD CREDIT AND AN EXCLUSIVE GIFT!
If you make your Disney Cruise Line reservation with Dreams Unlimited Travel you’ll receive these incredible shipboard credits to spend on your cruise!