I’d first attach the SSD to the old computer (additionally to the system drive).
Then perform the plex data transfer using this guide: Moving Metadata on Windows
Then end Plex and export the Plex registry data, as explained in the guide https://support.plex.tv/articles/201370363-move-an-install-to-another-system/
From now on, Plex server must never be running again on the old machine!
(if you still want to use it as a ‘fall back’ server, you need to to some registry tweaks - but more about that later, if you need it.)
Then install Plex server on the new machine
install the SSD (make sure it has the same drive letter it had on the other machine)
make sure all the media files do also have the same drive letters and folder names like before
then shut down plex server
and import the registry data
startup plex server again
