The only place I got wet in San Antonio SW was on a log raft ride (people can stand along the side and shoot you with water guns, and there are a few splash spots and a waterfall.) I didn't see anyone in a bathing suit... Most of the rides were dry.
I would recommend wearing lightweight clothes that will dry quickly if you do get wet. Slacks and shorts, light t-shirt... Maybe roll up an extra outfit for each family member and put in a ziplock bag, so if you get wet and it is uncomfortable you can change. (Have an extra bag for wet clothes...)
Food and drinks are prettyexpensive, so a backpack with water bottles and snacks would be a good idea too. (You can use that to carry your clothes...)
And don't forget the sunscreen. ;)