John I have exactly the same problem. Fortunately I have 2 identical cameras. I am able to swap components to isolate the root cause. It's the on/off button, not working! Your camera will actually function in all modes, without a battery inserted. Provided the charger is connected. It's my betting that yours wont turn on if you remove the battery. Thus, the charger is by-passing the switch, as it should, in order to re-charge the battery. But the power to the on/off button via battery and/or charger is not connecting. Sorry, I cant offer good news. If I knew how to fix the problem, I would fix mine and tell you the repair method. The internals are microscopic. I can barely see to trace the tiny PCB's. But if any moisture is inside, you open the camera door, remove components and pop it in a really warm place for a few days (airing cupboard-NOT the oven!). These cameras have sensors which protect from condensation damage. If your lucky it may be simply override protection.
John Mitchell, July 2009