The Idea is to CHECK your AC adapter to see if it is getting warm/ hot. If so, it is an indicator your battery is dead. But from what you have said already.. buy a new battery.
They don't take to being dead for any length of time, it has the effect of making them un-rechargebale. You can run on AC only , and I would if I were you. Pull out the battery and while your new one arrives, run it that way. A new factory battery is vastly to expensive. Go to Ebay and buy one there, from a trusted seller. Just make sure it has the same or slightly higher mAH rating, at the same voltage.