What you're describing is a slower shutter speed due to not enough light. You'll have a problem with any camera because it will have to use a flash indoors, which means it'll take much longer from shot to shot so that flash rebuilds it's charge, and the reason it takes a little bit to focus fast is because indoors there's not a whole lot of contrast and it takes a little bit longer to find a focus point.
The reason your old camera would do a pre-flash was to light up the scene so the auto-focus could find something to lock focus onto.
What it really sounds like you want is the speed and low light ability of a DSLR, with a large aperture lens, in the size and price of a budget compact, which you'll be disappointed with any of them if you can't accept their limitations.
I would love a compact camera that indoors was fast to focus, compact, cheap, and had good high ISO capability to be able to get shutter speeds that don't show a lot of motion blur but they haven't invented that camera yet. The closest compact camera that fulfills some of your needs is the Canon S90, but that puts you close to $400 and doesn't fulfill many of the things that you want.
If I were just making a choice of cameras then I'd go with either the Canon SD1400 or the Sony W350. Just go try both of them out and see which one you like, but don't expect them to do the things you want.
I am looking for a point and shoot that is fast to focus and fast to the next pic and doesnt blur when my kids move in the middle of the pic.
I have had Fuli's, Sony's and they never were able to do the job quite as well. Granted they were older models. the last Sony i had was the 7Mega pixel Sony Cybershot W??? I got a couple of years ago. IT was ok but lagged a bit as well. because of the 50 flashes it would give off before taking the shot.
I need SPEED and CLARITY!!!
SO do i get a CANON SD1400 an A3100 OR do i go for the SONY W350/370 - DSCTX1