Okay, I removed the code that I followed from adding local fonts.
I have added the below code to my functions.php, which is what I had tried initially without success. I went back to that, and retested and found that it DID change the font to Limelight for the page title and the menu, but not for the headings. The headings were what I was trying to change in the first place.
Limelight doesn’t have many variants as far as I can tell, so don’t know if I have the right thing there.