I just did the Freycinet circuit with my kids over three days on the Easter weekend. We started walking on the Saturday and the day before there was quite a lot of rain. Nevertheless there was no running water in any of the creeks we crossed and not even any puddles apart from a bit of mud in one of them.
We didn't get to the Wineglass Bay campsite (walked from Cooks, across Isthmus to Nth Wineglass and then out all in one day). We carried enough water to last us for three days, because we knew the water was unreliable, and that even if there was water in the Cooks tanks, we may not get that far (with kids as young as 5 we were prepared to stop and camp at Hazards beach if necessary - although our 5 year old ended up doing a valiant 7.5 hours and got all the way to Cooks Beach on the first day without much complaining - I was very impressed - walking with another family of kids gave them the extra motivation they needed, I think).
Instead of carrying water to cook with, we carried all fresh food (nothing dried) and therefore did not need any water to cook with. Of course this meant heavier food, but we did not waste weight carrying cooking water at all.
Turns out there was plenty of water in the Cooks beach water tanks last week (lots of mosquito wrigglies in there too, but easy to filter out by pouring your water from one pot to another).
We walked in one day, and out on the third day. On the second day, we did the easy walk from Cooks to Bryans Beach and back again, with a lovely lunch at Bryans with the kids playing on the beach for a couple of hours. Fantastic walk - apart from carrying soooo much water (which I don't usually do) and two tents and three sleeping bags. The price of walking with very young kids.