OK, I just realized what the problem is. First, I was wrong about a couple things. The only tag it's not adding is the title tag. The others are fine. So the two problems are that it's not adding the title tag and also not changing the name of the file to include the track number at the beginning. The second thing I was wrong about was that this problem has apparently been happening for quite a while. I just hadn't noticed it until recently.
What I finally discovered is that this problem is only happening with songs I have not changed the titles of in the Info window. If I change the title field in the info window, it will fill in the title tag and add the track number to the file name just fine. If I leave the title unchanged and only fill in other info fields (including the track number), the title tag will not be added and the file name won't change.
The way to fix this would then be to go back and find every file name in the iTunes Media folder that doesn't have a track number at the beginning, go into iTunes and change the title name in the Info window to something else and then change it back. Needless to say, this is going to be a real pain to do! I wish it would simply add the title tag and change the file name when I simply add the track number field in the info window.
Did that all make sense? Can you think of an easier way to fix this other than manually changing the info twice for each of the songs?