Reading of your children, thinking of my 2 grandchildren and my first digital superzoom, a Sony H5, reminded me of their biggest shortcoming- shutterlag!
It's no problem outside, when you can simply set it to "Sport" and get near instant shuter response, but inside, with moving children it's a real headache! And "Sport" doesn't seem to work inside, at east on my camera. When trying, the LCD screen is very dark, and I do think that's normal, but the only other choice is to use another setting and flash to freeze- but there is still shutterlag. You point and click, but the child is 3 feet away when the flash goes off. Using any high ISO feature is begging for disappointment.
Any DSLR will avoid you this lag- but, as you know, comes with a pricetag. I now carry Sony's great a700 for most photos, but still continue to use my H5 when wanting a lighter load, smaller bag. And I agree with Bob and his choice of all 'round lens- my Sony 18-200 f3.5 tele. By far my favorite and most used- though I have a couple, sharper, more specialized and thus limited lenses.
I have been reading more and more of Fuji's smart point & shoots. I believe you should check out their $300 and up models. Do handle and test one first, ensuring they've licked the shutterlag problem- indoors. Out of doors doesn't seem to be an issue, again reminding you of the "Sport" setting. I have 2 superzooms, both Sony, and both suffering from this "way too dark LCD" when Sport is used indoors. Can I assume yours gives the same problem? So dark you can hardly make out which child you're shooting?
The other settings, no matter what shutter speed you may select, still hesitate prior to capture. I bet that's the crux of your problem. All early point and shoots share that biggest of problems. But if your shooting is mainly your kids, your annual vacation and holidays, I'd say shop around and Make Sure you get a $250-500 P&S offering No Shutter Lag indoors, without flash. As partial as I am to Sony I would research Fuji, often offering the best camera per dollar, to include a great feature set. Good Luck! I envy you set to buy a new camera!