If you go to the playlist section of your app, there will be a three bar symbol in the corner. This allows you to customize your list to appear ony certain albums ad you or someone else might have clicked this. If there are any options on, change the options to off or none and you should be able to see yourplaylists again.
Now for the part, that you can't make a new playlist, you might wanna delete and redownload the app or icloud reset the phone if it matters that much. I hope that you do because it is a really quick and easy process to do and spotify is honestly worth it.
The “three bar” symbol allows me to pick music that Spotify thinks I want to hear. It does not allow me to access the music I have already saved to use. I cannot see the music “I want”. I can only see what Spotify “suggests”...mostly shot music. Again I hate it