Too bad you won't go with a fanny pack. That's exactly what I'd recommend. Anything you have to carry has to be left behind as you get on rides. A small fanny pack can stay in back or be twisted around to sit in front. No problem on rides - I've never been asked to take one off. I'm not talking about the mega-sized ones that hold two one-liter water bottles though. Just get a small one for each family member (maybe not the kids if they're real young). Drop in a small water bottle that you'll be refilling often, some Tylenol/Advil and bandaids, sunblock, sunglasses, a little snack ... and you're probably good to go. I'm not sure that PSK's and FAK's are all that necessary inside the park.

A backpack or carry bag will not be allowed to stay with you all the time. If you don't mind leaving it to be picked up after the ride, I guess that's fine. I would consider it a hassle myself. If you're dealing with very young kids however, you'll be needing to carry lots more stuff than I'm thinking about for my situation.