Jack,
What you are experiencing is the Shutter Speed-ISO-Aperture triangle. Higher ISO and wider (lower number) aperture can give you a faster shutter speed which is what you need. Indoor and low light photography is challenging and either requires a powerful flash or a good lens that will allow in more light.
In your photos, the first one was shot at 1/5 sec. at f/4 and ISO 1600. Second shot was at 1/10s f/4 and ISO 400. In both shots, the moving subjects were blurry because of a slow shutter speed. You would need at around 1/100 of a second to get a cleaner shot. In your case, you were already at ISO 1600 f/4 in the first shot. f/4 is the maximum aperture you could achieve at the zoom level you were using so there isn't much else you could do for the first one other than turning on the flash. The second shot would benefit from a higher ISO setting which would allow the shutter to freeze the action better. Also, shoot at wide angle so your lens gets the widest (lowest number) aperture it can use; when indoors, you are not going to zoom in much anyway so you should have plenty of pixels to give you a decent photo if you crop out what you don't need.
When I shoot in low light situations where I cannot use flash, I use high ISO and deal with the graininess of the photo. I always say that I can fix graininess in post processing but I cannot fix blurriness. But there are some situations where I know I just cannot get the shot so I do the best I (or my camera) can.
Incidentally, I use a Lumix LX3 as my point and shoot which has a good lens down to f/2 wide open and zoomed out. My full size SLR is a Canon 5D.
Camera geek signature:
Canon 5D, 24-105 f/4L IS, 70-200 f/4L IS, 70-300 f/4.5-5.6 DO IS, 50 f/1.8 II, Speedlite 430EX II, B+W Filters