You can try the following to rule out a software problem:
Reset iPod touch: Press and hold the On/Off Sleep/Wake button and the Home
button at the same time for at least ten seconds, until the Apple logo appears.
- Restore the iPod from backup
- Restore the iPod to factory defaults
Apple will replace the batry for:
Here is one third-party place. Google for more.
http://www.digiexpress.us/biPadb_c_42.html?gclid=CLOUu5T106UCFcbc4AodwW3UlAYou can also replace it yourself. See the following for instructions and where to get the parts.