BBB = we brought a fairly good amount of cash (I prefer to shop in Mexico with cash -- too many surprises in the past on credit card bills with conversion rate, etc), and a large number of travelers' checks.
Edited to add: I left all the cash, the TC's, my jewelry, wallet, DH's wallet, special pins purchased at MK before cruise, etc. in the safe, and never thought a minute about its safety. I used an old AAA card to secure the safe, and hid that deep in some clothing elsewhere in the room. I gave Guest services my Disney Visa card number at check in, and every once in a while as I walked by Guest Services stopped by and got a print out of my account if they were not busy. I don't like waiting in lines either.
The charges to my account when I checked it were all correct, so I let it go to the end of the cruise.
I would note one thing. I would also not like to give a company carte blanche to my debit card. From your counter, you have more than enough time to get a either a Disney Visa or a miles card -- believe me, this is the way to go! If you pay your bills online, you can set yourself a budget for onboard expenses, leave that amount in your checking account, and have a separate online payment made the last day of your cruise to your card. That way, you earn something from those dollars (ours were quite a bit more than I had expected), and not worry about your debit card. I also felt more comfortable shopping and dining at various ports with a credit card or cash, and not my debit card.
I used my Disney dollars at the Disney Memories store in DTD on the last day of our cruise - we had about 5 hours before our flight and drove to DTD. I was thrilled to be able to get what I wanted to scrap my cruise, and not have to worry about how much I was spending -- it was free!!