Problem solved. I got a hold of the repair manual, and it had a pretty good troubleshooting section for the horn. (By the way, an easy way to check if the horn relay is working is to crawl under while you hit the button. If the relay is working, you will hear it click.)
I tracked it all the way down to the horns themselves. Dead. A pair of 1996 Caprice A and F note horns off eBay, and I'm good to go!