Hi Casey,
I've just had a problem that sounds like the same thing. I changed host for a moodle site and the whole thing shat itself. All the images, pdfs, video, audio werent showing. Turns out the files were corrupt.
After some digging around I figured that Moodle creates a hashed version of the file you uploaded and saves it in a folder called filedir (or in subfolders of filedir) as something like this: 535fdcfafd0dd1ea35c108b4519d40de258e5f48
When we FTP'd everything across to the new host we transferred it as ASCII when apparently it should be binary. So we FTP'd the filedir folder across as binary and everything worked!
I used Filezilla for FTP. To changed to binary: Transfer > Transfer Type > Binary.
Hopefully this solve your problem too.