1996 Toyota Corolla Have cleaned battery terminals

I have cleande the battery posts and connectors. The head lights come on, so I am getting juice. I have taken the car to various shops and have had the alternator, and starter checked, they all pass the test. Yet upon turning off the car it doesn't seem to want to re--start. While running if I remove the negative terminal then turn on the headlights the car dies. Immediately upon turning the vehicle off, and attempting re-start in makes the click, click, click, sound as if the battery is dead. Battery is at 12.5 under no load, and immediately upon puttig jumpers on it, she fires up like nothing is wrong?