Bacon, ESTA can only be done online via one particular site (there are lots of con sites that charge you extra!) This is the official site you need to use. It's $14 per person and used to be very easy but people say it's quite confusing now, especially when applying on behalf of children. I've not done the new form but Google something like 'children new ESTA form' and that should help.
Ruggles, welcome! I usually do barefoot at waterparks and you'll prob be fine at this time of year but the floors can get extremely hot later int he year. Crocs are very comfy but I can't do them in the rain (of which there is a lot in Florida). I used to have serious shoe stress but I bought these walking sandals and they're amazing. I think proper walking shoes/sandals are the way to go - some are less ugly than others.
As for what to wear in waterparks, most people wear a costume all day with some sort of coverup when you go for lunch etc. I don't think you'll want to wear shorts because your costume will stay wet for most of the day. Loads of people walk around in just swimming costume all day so if you're comfortable with that, that'd be absolutely fine - personally I'm only happy in a costume when underwater :)