You could export the playlists using ExportTools
You can use WebTools to export and import playlists
(both of the above are still working, if you update to the latest releases of these plugins!)
The most seamless experience though, would be to copy over the Plex data folder from the old to the new server. And keeping all storage locations of media the same (if possible). This will retain all ‘watched/unwatched’ status and previous playback history and manual metadata edits etc.
Changes to the media storage locations can be made by editing the database directly (recommended only for advanced users): [HowTo] Plex database modification - Moving media the right/wrong way
