I recommend a small backpack or a comfortable cross body bag if you don’t need to carry much but just avoid the bags that have drawstrings/ties I had the Disney parks one before and it gets very annoying compared to bags/purses with just zippers IMO. We stopped bringing water bottles and just get free cups of water at the QS spots.
We used a Burton Prospect Pack for the last two trips and it was incredible. It’s roomy for a smaller backpack, fits under a stroller and has a chest strap
Never had a problem on Splash Mountain, my bag always sits between my ankles and has never become wet. On Kali River Rapids at AK you have to store all your belongings in a locker before you can ride that one.