Camera thinks plugging in means turn on?

(My camera's actually an M552 but that wasn't an option on here)

Every time I plug in my camera, the light between the power and photo buttons blinks once or twice like it's supposed to; after a second it stays on, then the backlight on the screen turns on. After a few more seconds the camera just turns on and there's no "plugged in" indicator on the display.

If I turn it off when this happens, it turns off and acts like it isn't plugged in to the charger at all. Sometimes moving around the cord will make it all happen again, as if the cord were out of alignment.

When it's plugged in the battery seems to be full, but unplugging it reveals it to be critically low.

I've tried it with at least four different charging cords, outlets all over my house, and I've tried just leaving it plugged in. It simply will not charge like it's supposed to. I've tried cleaning the battery and the battery socket as well as the micro-USB port.

Does it just need a new battery?