There are a couple possible things that might be going on here.
First, for your particular camera, when you zoom in, your aperture gets smaller, from f/2.8 to f/4.4. Because the aperture gets smaller, the camera has to use a slower shutter speed to let more light into the camera. It is usually very difficult to hand-hold cameras at slower shutter speeds. You get a lot of camera shake during slow shutter speeds, even though you might be holding it as steady as humanly possible and even though your camera might already have "image stabilization."
So this "smaller aperture while zooming in" effect might factor into this.
The other thing that might be going on is that when you zoom in, your minimum focusing distance is greater. When you're not zoomed in, your minimum focusing distance is 30 cm (about 1 foot). When you *are* zoomed in, your minimum focusing distance is 2 meters (about
6.5 feet).
So, if you're closer than the minimum focusing distance, then your subject will be blurry. I have a feeling this might be what's going on.
One thing you *can* do for these close-up zoom photos is set your camera to Macro mode. This will reduce your zoomed-in minimum focusing distance to 1 meter (about 3 feet).
I got the specs for your camera at:
http://www.imaging-resource.com/PRODS/FZ35/FZ35A.HTM#. It provides all the above numbers for your camera.
See if that's the solution to your blurry, zoomed-in photos.