Why can't my iPod nano work off the charger?

if your radio plug in charges it as well, and you keep it plugged in a lot it sounds like you killed your battery. I wouldnt suggest trying to replace the battery, getting into the nano's without breaking something is more trouble than its worth. here's a link to the battery replacement guide (the guide even says it is Very Difficult):

iPod Nano 3rd Generation Battery Replacement

If you decided to fix it or get a new one, you need to fully charge and then fully drain the battery every time to get the maximum life out of your battery. Leaving your ipod, laptop, iphone(or cell phone) or pretty much any electronic device plugged into a power source all the time is a terrible idea, it will kill your battery years before its lifespan.