The problem with Plesk is that it creates a parallel configuration hierarchy to the standard OS one. Everything that normally lives in
/var that's related to the Plesk-provided services gets mirrored somewhere under
/usr/local[*]. Plesk only pays attention to its copy of the files.
Normal backup and restore strategies for a CentOS box therefore won't help you.
Plesk being Plesk, the "right solution" to this problem is to use Plesk's own backup and restore mechanisms, rather than try to go behind its back and play with the filesystem directly. There's a whole chapter on this in Plesk's user manual. If you had a backup, you could just restore on the new machine. (Ideally, anyway.)
If there truly is no way to make Plesk on the old machine give you a backup, you could try finding that
/usr/local directory I'm referring to, and just try to mirror that tree, but I suspect all you will accomplish with that is to transfer your problem to the new machine. For that matter, I suspect a Plesk backup taken through the GUI at this stage would probably transfer the problem, too.
A better long-term plan is to learn enough about CentOS and Linux in general that you can move off Plesk onto a generic unmanaged VPS. When you build a server up from pieces yourself, you implicitly learn what has to be backed up, and how to restore it.
[*] I forget the details, having moved off Plesk years ago.