Given your requirements, I would second Corpsy's recommendation of the Fuji S6000FD (S6500FD in some countries).
The reason why photos of your kids appear to be blurry is because you are using a slow shutter speed. This problem is predominant in indoor situations as the camera has less light to deal with, and is forced to slow shutter speed in order to gain a decent exposure. IS will not help as it only stabilizes camera shake, and does not increase shutter speed. IS will not cancel blur caused by subject movement.
What you need to do is to increase shutter speed for indoor (lowlight) images. This can be done by increasing the camera image capturing sensor sensitivity (or ISO). The catch is, higher ISO's result in grainier images.
Fuji's S6000 probably has the best high ISO capability amongst the ultrazoom cameras, much better than the Canon S3 IS, and probably on par with some entry level DSLR's.