The ear jack could be faulty or it could be a problem with the docking port.
It sounds as though your iPod is in docking mode.
1) Clean out the docking port area carefully to make sure there is no dust or lint jammed in there.
Check for damaged pins, if you see bent pins then there isn't much you can do as the dock port is soldered to the main board and would cost you as much to replace the iPod as it would to fix the docking port.
2)If you are up to it, you can open the iPod and replace the ear jack. It is a small replaceable plug in jack so not a big job once you get inside.
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