I tried most of the ideas listed here to no avail. Finally, I realized that the songs that were not authorized (or able to be) had been purchased with the account I currently use when it had a different e-mail address as the Apple ID. I went into my Account in the iTunes store and changed the Apple ID back to what it used to be (my old e-mail address) and saved the changes. Then, I went back to a song that wouldn't play and attempted to authorize; it worked. I then went back to my Account on the iTunes store and changed the Apple ID back to my current e-mail address. Everything seems to be playing fine; the older songs are authorized and the more recently purchased ones are as well.
I don't know if my method got rid of a symptom of the problem or actually solved it, but all my songs are playing perfectly fine now, so I thought I would share. Hopefully this won't happen again--I'd rather not have to keep changing my Apple ID back and forth.