We did this on our last trip, but we also had a stroller. I used a soft-sided cooler (a lunchbox size) and put 6- 20 oz. water bottles inside. I filled two qt. size ziplock freezer bags with ice and put on the top and bottom of the cooler. (I also filled my reusable tumbler with ice water prior to leaving for the parks.)
Whenever QS water was convenient (snack or lunch stop), we topped off my reusable tumbler. When it wasn't, we used the water bottles we brought in. By late afternoon, the ziplock bags of ice had melted but were still cold. We poured those into the reusable tumbler and drank that too.
Next time, we will not have a stroller. We have only taken one backpack in the past, but I can't imagine packing 6-8 20 oz. bottles of water in one backpack, even if we keep switching it off. I think we might try with 2 backpacks, but I still haven't decided. I'm also entertaining the idea of getting camelbacks (or something similar).