Hi James,
Without seeing the pictures (plus the exif details) it's difficult to know what the problem might be.
Hopefully it's not Jessops knowingly off-loading a bunch of duff cameras - I think that is extraordinarily unlikely. Of course yours might be duff, but there are plenty of other more likely possibilities to look at first.
I am no photographer, but from my digiscoping I know a little bit about ISOs, shutter speeds, aperture settings and so on. What I am quickly learning about the FZ20 (I've only had mine three weeks) is that this knowledge is invaluable. This is a very sophisticated camera in comparison to most consumer cams I've used up until now, and great fun... if you take the time to learn to use it.
I have taken more than a thousand shots in the last three weeks, many just aiming randomly about the garden on different settings - some even of the tv screen (!) just to play with different features. I have shot in black and white and colour, aperture priority, shutter priority and full manual. I have used all three automatic focus modes, plus manual focus, plus all three exposure evaluation modes, and I've used a tripod, a monopod, image stablisation and none. And I'm still only just getting to grips with things.
I would suggest spending some time with the manual, maybe some time looking at the dpreview panasonic forum (loads of great advice there) and much more time just taking pictures. Please don't get disheartened by the great shots you see online - remember, it's the photographer who gets the shots, not the camera, and there are some great photographers out there.
As to your specific problems, if you have got blurred shots it might be one or both of the following problem types:
1. Slow shutter speed leading to image shake - even with image stabilisation, a really slow shutter speed will show every tiny hand tremor (or subject movement). I can handhold down to about 1/30 (which is pretty good for any camera) - below that it's either time for the tripod or time to look at increasing the light coming in (wider (ie lower) aperture setting, higher ISO sensitivity (100, 200 or 400 - I prefer to shoot in 80 if I can), flash (as long as it's within range), or improved ambient lighting (where practical)).
2. Incorrect focusing - often causing by using the wrong auto focus setting. The autofocus is pretty good but not infallible, particularly if you've not got the most appropriate mode selected. Try changing it to spot focus rather than multi-focus for difficult subjects, or use manual focus (which I find incredibly helpful when focusing on a bird feeder for example - just focus once and leave).
I hope this helps and you aren't too disheartened - just remember, you've got a fantastic versatile camera that should improve your photography no end, and all at a low price that has the rest of us absolutely green with envy.
Good luck.