I have not read this entire thread but what one thing I learned from some of the experts here is to start with the simple possibilities first. Did you check the path to your attachments directory?

Go to attachments/storage type and check the path and change to your new server if you have not checked this.

You restored the site on this new server though, right? If your mysql is different on both servers, then restoring the backup isn't going to change things. There is something different between the two servers that is causing the issue.

Have you run Maintenance > Diagnostics > Suspect File Version?

Have you turned on Show Invisible Characters when editing the text file to verify a BOM wasn't added?

Hi Lynne.

Suspect File version brings up:

class_core.php File does not contain expected contents
functions.php File does not contain expected contents
init.php File does not contain expected contents

Replaced those files using original vBulletin files and my attachments are now working.

Thank you to both yourself and kh99 for helping me with this issue! I really appreciate it :)

I'm glad you got it working. :)