After much trial and error, here's the simplest and only reliable way I've found to copy entire playlists from one Spotify account to another without having to "follow" the owner of the original playlist.
NOTE: This method only works using the Spotify desktop client installed to your local machine. So wherever “Spotify” is referenced below, you must perform that step using this client (as opposed to the Spotify web site or mobile app).
First, set up your directories:
Next, grab the links to the playlists you wish to copy from the OLD Spotify account:
Next, copy your playlists over to the NEW Spotify account: