[AZKen]

Jake, I can't tell if you are testing the horns without the relay or with the relay or both. Anyway, first test without relay. Assuming you only have ONE contact on the horn: Supply hot to that contact and touch bracket or case of horn to ground. It will blow. That's got to happen first. You can use the battery posts for this test if you wish. Hook a hot jumper to the horn contact and touch horn case to the battery neg post.

Then, With relay: If it is a three terminal relay, one terminal is hot all the time, another goes to horn and the third is grounded by the horn button. When the horn button grounds at the steering wheel it closes the relay contact and the hot "all the time" terminal is internally switched to the other terminal that goes to the horn (that's the click) and thusly supplies hot to horn terminal. Honk! Honk!
If your relay is incorporated into a new fuse box, then you will probably have instructions on wiring to horn and to horn button, etc.

The horn itself is grounded by it's mounting bracket.
So identify which terminals are which on the relay. You can test the relay with a 12V bulb to see if it works. Touching the ground terminal of the relay to ground is a way of simulating what the horn button does. THE CASE/BRACKET OF THE HORN MUST BE WELL GROUNDED when testing horn. Clean shinny metal, no paint, no plating, no insulator.
Just as a side note: Horns have a screw for vibration/sound adjustment to get the "note" you like. I guess it's possible that screw is too tight or too loose and horn vibrator is "locked" or "detached". But on two horns?

Repost if you need further.