How do I move my PSE catalog and picture files to a new computer?

I'm afraid it hasn't worked.  I will be seeking out further support from Adobe and will let you know if I get any better resolution.

Unfortunately, no.  I just tried to perform a similar operation by adding a hard drive to my computer because my old drive has run out of storage space. The problem is that the Organizer recognizes your files by the hard drive's Volume ID.  This is a low level identifier which can be changed if you know what you are doing and are willing to use old DOS commands in a Command window.  (I'm probably going to try this myself today and if you are interested, I will let you know how it goes.)  Simply changing the Drive letter doesn't work.

The recommended method for switching computers is to make a backup on your old computer and restore it to the new computer.  See this article for full instructions.  You can make the backup in PSE 12 on the old computer, and use PSE 2025 on the new computer to restore the catalog.  The catalog will be converted to the Elements 2025 format as part of the restore process.

Before performing the backup, I highly recommend that you Find>All Missing Files from the Organizer's Find menu, and either reconnect or delete them.  It may also be worthwhile to perform a Repair and Optimize from the File>Manage Catalogs dialog.