...so when you put them on your iPod, the songs go into the artist folder and not "Various artists"?

I think you have to go into "Get Info" and change where it says "Album Artist".

I have a lot of soundtracks, and I like to keep them together, so I have the artist who the artist is, but under "Various artists". Previously, when I just had the artist, it went under the artist's folder, but now it doesn't seem to.

I have the same problem even if I change the Album Artist to "Various Artists"

If I uncheck "Part of a Compilation", do you think it could work?

Nope. Totally didn't work. Just made it so the artist's name appears under the song title.