when the AC is connected it says the battery is fully charged
Yes, I've noticed this as well, it's annoying (on top of that the orange charging light never turns green).
and won't charge the battery
No, I have not noticed such a phenomenon, logged-in both as root and as a normal user. My batteries (EA-BL06 and EA-BL08) do get charged. I am charging via the USB cable.
Maybe you need to wait a little bit longer as charging takes quite a bit of time especially with screen backlight set on high (5 or 6). Try also first suspending and then charging to see what happens. I assume you are using the proper AC adapter for your country (and not using, say, the 100V Japanese one in the US)?