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: How to back up
- 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. The appointment is free.
Apple Retail Store - Genius Bar
Try a third-party place like the following./ 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