Battery apparently charged, but will only run when plugged in. Why?

I think this is a battery problem, but as I don't have a spare battery, I can't really test it. I wanted to see if anyone else has had this problem or a solution.

My battery is (allegedly) charged. The light on the adapter glows green, the battery icon claims 100%, even coconutbattery says it's 100%, but when I unplug the adapter, the computer shuts off. It's as if there's no battery in at all.

I was having problems before, and I thought it was only the battery, so I bought a new one. It helped a little bit, but then I realized there were other problems. (DC-in board needed to be replaced, plus there was a crack in the bottom case that made for a very loose connection to the battery. Unless I pushed in the right spot, the computer wouldn't recognize it was there.)

I've replaced the bottom case and the dc-in board, so that leads me to believe the battery was just junk in the first place. Coconut battery claims I've only charged it five times, which sounds accurate.

Any ideas? Reset PMU? Try to find a spare battery and test it out?

Oh, other little side note, which probably has no significance: If I disconnect and reconnect the battery, the light will switch to amber for a few minutes, though the icon on the screen will still claim it's completely charged.