Hey Lauren,
It's strange that the camera does not work when the camera is plugged in, though some devices are programmed like this. Your best bet is probably trying it with a replacement battery; they shouldn't cost very much money.
If you don't want to spend money quite yet, you can open up the camera and check all of the connections to the [guide|5004|motherboard]. Having not used the device in awhile, one of the connections may have come loose or gotten dirt or something stuck in it. It's a pretty easy device to take apart.
Also, you can check the battery connections for any signs of corrosion. If you find any, you can clean it with a cotton swab soaked in CLR or vinegar.
Hopefully this helps!