I know how difficult it is. From the description, I don't think the OP's problem is blurred photos, but rather a problem with shutter lag. We have to take a lot of pictures of our animals for use on our website. The babies are the toughest to photograph because the "kodak moment" is really just that - a moment. And, especially younger animals don't like "posing" for the camera. Unfortunately, the shutter lag = "moment + 0.1 seconds." Of course, one of the cameras we use is much better than the others, but it seems the biggest problem is the time it takes to automatically focus. I'll leave it up to others to make specific recommendations.
However, here are some tips I've found, having taken thousands of shots of animals:
1. I recommend taking the photos outdoors where the natural light is better; and many cameras auto-focus faster with the brighter light.
2. On many cameras, if you hold the shutter button down halfway while you're the distance away at which you're taking your picture, it will pre-focus. Then, at that moment, press it the rest of the way.
3. Take lots of pictures of each animal; pretend it's like one of those shoots of models where the camera is just taking picture upon picture upon picture. Look at all the pictures afterward & pick the best photo; don't just rely on 1 or 2 shots.
edit: by "at that moment" in #2, I don't mean the moment the camera is focused; I mean the moment when the animal is posed just right.
Also, fwiw, I had an older Fuji camera that was quite fast; I think it was only a 3MP camera. Rather than looking at the brand new market for a faster camera, you might want to take a look at the used market (particularly since you said animal shelter & I know they're not exactly rolling in the dough.)