There's dehydrated water everywhere!
We do carry water. At Clarno there is water at the parking lot, and at the bottom of the palisades trail as well. We carry water for ourselves and the dog in our rig; I refill everything every time we stop anywhere, and when we get out to hike, I encourage Klaus to play in the water (I spray him) so his ears and feet get wet.
I also try to avoid hiking in the true heat of the day, between 11-4, if at all possible. Though honestly, I don't mind the heat. The kids wear hats. I do too if I fear I might fry my brain. We go pretty slow to accomodate my son who likes to dink around, and we take a few minutes whenever we come across shade.
Yesterday we did the first part of Clarno (the..ahem...no-dog hike) around 10:30; and subsequently we didn't go too far. We hiked in maybe an hour or so; that's not very far with little kids and a young dog. I really didn't want to be walking back at noon, at that point the day looked as if it might get really hot.
It didn't, though. It got up to the upper 70s, which was plenty warm but not truly deadly. Just warm enough to get the reptiles going.