My laptop (Toshiba Satellite Pro A300) shuts off as soon as the ac adapter is plugged in. Sometimes, though, I can get 15 minutes. HOWEVER, it doesn't do it in safe mode. I have updated the BIOS, re-installed the OS, cleaned any dust out, tested with another ac adapter.
How is your power/charge setup ? Does it allow you to set to shutdown for recharge ?
Check the settings on how the laptop is set to behave when plugged in.
What does it do if you remove the battery, plug it in, then turn it on ?