I just tried to power with a MacPro and also tethered through C1 with laptop, battery still draining.
Did you update to new firmware 2.0? If yes, then I'm going to presume problem is just my body.
No experience with the supplied AC adapter, but I did inadvertently leave the cam ON and exhausted the battery, by accident, during a day shoot.
Plugged in a Mophie battery powerpack (Powerstation XL, 12000 mAh), via a standard USB cable, and the cam resumed full operation within seconds, Note: you can't power ON with the bat-pack and charge the battery at the same time. Switching the cam off, though, and the external battery pack switches into charging (the camera's internal battery) mode.
I have and do tether with C1 via an MBA - but have no first hand experience of doing that with a depleted battery. Like geesbert, have also used a mains dummy battery AC power supply (ExPro) and had no problems.
From what you say, and my experience, it seems the issue may not be the cam but the supplied AC/USB charger. Can you try eliminating it by using an external power source (either or both of the two solutions above) and see what happens ?
Edit:
< I just tried to power with a MacPro and also tethered through C1 with laptop, battery still draining. > Not 100% sure, but don't think that you can tether AND charge at the same time .. ?