My iPhone 3GS is charging through wall plug but doesn't increase battery life?

It was knocked out of someone stepping through the cord while charging, now it doesn't charge anymore. Before that I had to recharge the phone every time. When I'm on text, or Internet it shuts down and reads 50% battery life or 12%. I recently deleted the games on my iPhone and transferred the games to my iPad2.To save battery life from draining. Now I'm at the point, where I have to replace the battery myself. Because Apple or AT&T won't help me with the cellphone. So I can use it for phone calls and texting. What's my next move of what I should do? Should I have the cellphone inspected for damage?

Added (1). Other actions I tried reseted the settings, managed notifications to important ones.

Yes you should get it inspected. I have a feeling that the battery is damaged or the software has a bug and is causing this. (i remember when IOS 5.0 was released iPhone 4 owners had massive Battery drainage then apple updated the software to fix the bug) you should try restoring your iPhone or get it checked.