iPod skipping multiple songs

Hey, so I have an iPod touch 5th generation running iOS 7 and it didn't start until today as I was trying to put a song on my iPod, but as I was trying to, all that would happen was the little spinning syncing arrows would appear but the song would never appear. After fooling with it for a while, I gave up and now whenever I try to play music, when I choose a song, it skips that song and then very quickly skips a bunch of songs - anywhere between 2 or 3 and 15 or 25 or more - and then the first song I clicked on becomes greyed out and unplayable at all and has a little downloading circle next to it. I've also noticed a pattern, that all of the songs that it will play after skipping are always without a doubt songs that I've had the longest, at around a year or so. It absolutely does not play any songs that I've added within the recent few months. I don't know if that means anything though. Also, not all of my music is on my computer and I didn't purchase it all with iTunes so I really don't want to have to restore my iPod. I also don't sync my music, I just click and drag the songs or album after I download them, so I don't know how or what un-syncing and syncing does. I did reset my iPod but it didn't help. This is really annoying me and I really want to fix it, so any help would be much appreciated.

Thanks.