First, confirm your songs play without skipping on your computer, when you play them using iTunes.
On the iPod, if you play songs while it is connected to your computer or a USB power adapter for power, do the songs still skip? To make it a good test, do a Reset (restart) on iPod
Learn how to reset your iPod - Apple Support
Then, connect it to the computer or a USB power adapter. If the connection is to the computer, eject the iPod in iTunes but leave it connected. Try playing songs while the iPod is getting power from the dock connection. Is there a difference compared to the iPod running from battery power?