Try:
- Reset the iOS device. Nothing will be lost
Reset iOS device: Hold down the On/Off button and the Home button at the same time for at
least ten seconds, until the Apple logo appears.
- Restore from backup. See:
iOS: Back up and restore your iOS device with iCloud or iTunes
- Restore to factory settings/new iOS device.
If still problem, make an appointment at the Genius Bar of an Apple store since it appears you have a hardware problem.
Apple Retail Store - Genius Bar
Apple will exchange your iPod for a refurbished one for $199 for 64 GB 4G and $99 for the other 4Gs. They do not fix yours.
Apple - iPod Repair price
A third-party place like the following maybe less. Google for more.
iPhone Repair, Service & Parts: iPod Touch, iPad, MacBook Pro Screens
Fix it yourself if you are up to it
iPod Touch Repair – iFixit