I normally just use my debit card for everything but it's carried in a wallet, in my purse, which I'll have neither of in the parks.
You will see plenty of people with bags (that likely have wallets in them) and even purses. I've been carrying one of my
Disneyland purses every day in the parks with me (Disneyland trip) this week. You just trim down what you normally carry. I have to carry sunscreen, Rescue Remedy, my camera, etc with me anyway, so a purse works great for that.
If you want a small wallet, the sideswipe wallet (just google it) is terrific.
If you want, get a nice lanyard (I'm partial to my Ravenclaw "school tie" style lanyard) and buy a hard plastic waterproof case to put cards/cash in. I use a clear Witz-brand one; they are called See It Safe. I got mine at REI.
Even when I choose to bring a magic band in, even if I choose to set up room charging (which is rare, and generally only when the resort has messed up and has decided to give us their typical $200 on the account, which FORCES you to set up room charging), I bring:
driver's license (or passport, currently, as I lost my wallet and am waiting on my replacement DL to arrive)
CC
debit card (Disney Chase)
Ticket/AP depending on where I am and what I have
insurance card (just b/c the law says an ER has to treat doesn't mean they won't come in every 36 seconds asking you or your companions for your insurance card!)
a bit of cash