Q: Why is my iPod touch not changing the wallpaper I have selected???

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.

- Reset all settings                  

Go to Settings > General > Reset and tap Reset All Settings.

All your preferences and settings are reset. Information (such as contacts and calendars) and media (such as songs and videos) aren’t affected.

- Restore from backup. See:                                               

Restore your device from an iCloud or iTunes backup - Apple Support

- Restore to factory settings/new iOS device.       

None of this has worked for me. I've tried resetting it, and I have a 5th gen. The warranty has expired, therefore it is probably crashing just so I have to buy a new one. I'm not impressed. I am reluctant on factory resetting it, because I have done that once already, when it completely died on me. It brought it back to life, but it freezes, and apps shut down on me, and it's frustrating. I hate losing all of my pictures and messages. If I were to buy another iPod, or possible and iPhone would there be a way to transfer everything from my current 5th gen iPod to that device?

To transfer data to a new device see:

Transfer content from your previous iOS device to your new iPhone, iPad, or iPod touch - Apple Support

Did you backup and restore from backup? From yo description it is not clear you did that.