Why is my psp unable to function without a charger on

So heres my story... In the beginning of this year I decided I wanted to start using my psp again and so I ordered a battery. It arrived this week and I came across this peculiar issue.

(This psp is not modified in any way shape or form..)

My psp cant open with the battery only. But it opens when the powersuppply is on. If I pull the plug from the psp it will shut itself off and wont turn on till I put the powersupply back on. If I try to update my system it will remind me that "the battery level is low. Connect the AC adaptor then wait for the battery to charge" This seemed to be the case of a bad battery, but...

When the psp is on with the battery it indicates that the battery is full..(even though it wont opperate if I pull the plug on it)

And If I open the psp with the charger and the battery on, and once the green light is indicated but before the screen turns on I pull the plug leaving it with only its battery it will opperate successfully. It will continue to opperate even if I put the power adaptor on and then pull the plug. But it wont last long since if I perform a system update, If I shut it down or the screen saver turns on it will turn off completely.. Please HELP! The ps store is closing soon and I need to get some games... Thank you in advance! :)