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
If you posted a screen shot taken with the iPod that indicates that it is not a screen problem