Hey Danny,
I was there a couple of weeks back. The creeks were nearly all dry, and I didn't check the tanks at Cooks, but there was water in both. Plenty of wrigglers and bits of vegetation in the tank at the back. I cleaned out as much possum poo and compost as I could from the gutter on that side, but I was too short to reach some of it. In contrast, the water in the other tank (which doesn't have a wattle tree near it), looked, tasted and acted clean.

Maybe you can reach more of that gutter? The hut was rather dirty when we got there - clearly hadn't been swept for a good while. Next time I'll take a bigger bag for other people's rubbish.
Don't leave ANYTHING in the hut overnight. A friend nearly lost her camera, which had been dragged off the table to the entrance of a rat hole, had the wrist strap chewed off, and I only noticed it as we were leaving (another good reason to sweep thoroughly)! Fortunately it was shockproof, and still in working order.
There was a little water in Eastern Creek iirc, and a trickle in the grotto where the track crosses Graham Creek. I wouldn't rely on them though. The ranger confidently told us there was water at Cooks, so we just went with that. I'd take puritabs or similar next time, just in case we needed other water. There was
I don't have pics, but heading from Cooks to Graham, there were a couple of tent sites right near the track, not long before the signed Mt Freycinet turn-off. One had a tree overhead too big for my comfort, but I vaguely remember thinking i might camp at the other one one day. I didn't allow time to climb the (out of date book) reportedly overgrown track up Freycinet. Now there's a spiffy blue sign inviting people along. I'll just have to go there again. *sigh*
