I didn't carry a purse on the ship, either. What you do need to have is your key card, so however you are comfortable carrying it is good. I kept mine in a secure pocket.
In the evenings, you might want a few dollar bills for tips for the dueling pianists. There is no requirement to tip them, and many don't, but if you really get into it and make a request or just really like the show, you might want to have them on hand. I didn't have any $$ the first night, but intentionally brough some the second night and put it in their tip bowl.
By the way, during Dueling Pianos, my sister and I asked for and received ice water instead of drinks. We just aren't big drinkers, and didn't want to spend the money. The waitpeople didn't even blink, and didn't seem bent on getting a tip out of us, so the second night I did tip with a little cash.
Nowhere else did we need anything else, though!
Have fun - travel light!!