If the playlists are in your new Spotify account, they should come over as part of your Spotify connection. If you’ve set them up using your old Spotify account, there is no easy way, as the pointers in the playlist point to the old Spotify locations. 
 

I’ve never used it, but I’ve seen reference to an app that I think is called soundlib, or maybe soundliz that has the ability to export playlists from one service account and import them into another. Which is what, essentially, you’d be doing.