What it seems you are having a problem with is getting fast enough shutter speeds to get rid of motion blur. What the camera needs is to kick up the ISO level in order to get a faster shutter speed so that introduces noise. The shutter speed does not influence the amount of noise/grain in a photo but ISO does.
So what you are wanting is a camera that is under $200 and takes non-grainy photos in high ISO settings. The only non-DSLR cameras that do that cost over $400.
Is there an inexpensive digital camera (under $200.00) that will take a decent NON-Grainy shot at fast shutter speed? I'm trying to get decent shots of puppies with a Canon A100IS with no success.