If it says Purchased AAC audio file, those songs do not use DRM. They are purchases from after early 2009, when Apple was allowed by content producers to stop using DRM for songs sold by iTunes Store. They are not the cause of authorization error message when syncing songs to iPod.
Restore would be a sensible thing to try next.
So, if most of those songs say Purchased, not Protected, about how many do you have that are Protected? If Music is the only media type being synced automatically, here's something you can try to figure out what specific songs are causing the error message... It could even be one rogue song.
Go to the iPod's Summary settings screen, and check the box for Sync only checked songs and videos. In your iTunes music library, show it sorted by Kind like before, select all the Protected songs at once, right-click the selection, and Uncheck Selection. After doing the Restore, set it up to automatically sync only Music. Choose to sync Entire music library (if your music library fits on iPod). Apply. iTunes syncs your iTunes music library to iPod, except for the unchecked songs (which are all the Protected songs).
Now, if you still get authorization error message, I don't know what's happening... If authorization message does not happen, checkmark a small number of the Protected songs, and sync again. By process of elimination (if the total number of Protected songs is not too large), see if you can find the specific Protected songs that cause authorization error message to come up.
Also, I don't think you mentioned your iPod model. In case it matters, what model iPod do you have?