I would not recommend any camera (under $850) for photographing active children indoors without a flash.
For indoor flash shots, the Sony S600 is a good choice.
It has a very bright flash that can cover a distance of 23 feet. And it will do as well as other cameras for those indoor shots without a flash.
Here is a review of the Sony S600:
As you found out, the Sony W7 does a poor job on any indoor shot. The Sony P100, P150 and P200 have the same problem.
I've been doing a lot of reading of camera reviews this past week since I dropped our Olympus C-4040 and it broke. We have a new baby and a toddler and need a new digital camera fast! Can someone recommend one for under $300? I tried a few in the store yesterday like the Sony DSC-S600 and Sony DSC-W7 - the first held up pretty well with my toddler walking/running by. But the W7 was shockingly blurry with and without flash!
--I really need a camera that's good for fast moving kids AND good indoors AND potentially without a flash indoors. That may sound crazy, but I hardly ever used the flash on my old Olympus (virtually all indoors) and it performed great! Almost never a blurry picture and only sometimes did I have to edit it and use a flashfill feature. THANKS!