Organizer: How to connect catalogues to a new device? (media already moved to new device)

Try this:

Set the organizer left folders panel in tree mode and select the 'offline' items in the bottom of the list.

Click on one of the thumbnails showing the 'offline' icon (red circle with vertical bar on top).

In the right folder in  Information mode, you have the 'General' tab in which you find a Location section. Click on Location to open a dialog offering to reconnect files or drives.

In the organizer, moving files externally, even with the same folder structure does not solve the issue of the drive or NAS  identification being different, so reconnecting is not a realistic option.

I don't use NASes myself with the organizer. Only disavantages for me. That does not mean that some users are not successful in using them, but I never see them coming for help in this forum. So, my suggestion is to use one of the classical ways to 'move' a library to another drive: use the left folder in 'tree' mode and move whole folder trees by drag and drop from the original location to the new external one. Be patient, that can take time. If moving a part of a folder tree works for you, you can take the time to move the rest.

Moving your library externally and hoping to reconnect like in Lightroom is at best very painful in PSE.