Update: I seem to have fixed the problem! I went to the "summary" section of my iPod after clicking on the device on the left side of iTunes. I checked the box that allowed me to "manually manage music and videos" and applied this option. The songs are syncing! And, yay, they're playing on my iPod now! Hopefully, this helps anyone else who might have this problem. There might be other issues, but this is working for now.