Is it my logic board?

I have a black 30gb 5th generation video Ipod. I'd been having trouble with it where it wouldn't charge when connected to the computer but would when I plugged it into the a/c adapter. Now it only works when it's plugged in but even though it says it's charging it immediately shuts off when you unplug it. It never says it's charged fully.

Is it the logic board?

Is it the battery?

My dad replaced the battery before giving it to me last year but I don't know if they can go that bad that quick.

Please let me know what you think so i can order the right part.