I’ve written some pretty clever and original song lyrics. Now what?
I realize that selling just lyrics alone (without music or a full demo) isn’t normally how it’s done. So, is there a trustworthy website that I can post my lyrics to, for other musicians to see and possibly license (or buy) them from me?
You won’t get anything from a display site except maybe requests for scammers. Musicians don’t have to search for lyrics; they always create their own, or know someone close instead of a stranger.
About all you can do is find some musicians, befriend them, and see if they want to write songs to your lyrics.
Maybe you could put an ad in the NME. It worked once, and pretty well.* 
Elton John and Bernie Taupin. Though that was done by the music company, not the artist.