As a small "update", I decided to try a little test, to see if perhaps my memory card was the problem. I downloaded some songs to the card, listened to them to hear them mess up (they did), then I cleared all my downloaded music and switched the storage space to my phone's internal memory, downloaded songs again and listened to them to see if they'd mess up, and once again, they did. So whatever the problem is, it doesn't seem like it has to do with the memory card.
I don't know what else to do, though... My home Internet is fine and works well with all my other devices, so I don't think it's a matter of having a bad connection for downloading. This is so frustrating, and I don't know what else to do...
Edit: Another update... In my searches for info, I saw someone mention something about having issues with music when headphones are plugged into the jack. So I tried downloading some songs and playing them on my phone's speaker, sans anything plugged into the jack, and lo and behold, I don't hear any issues with the music. That's extremely bizarre... There has to be a fix for this, right? Maybe? I hope...?