I recently bought a 2003 Heritage Softail and I'm trying to track down an electrical gremlin. The horn stays on. I checked the wiring going to the horn and it has continuity with and without the horn button depressed. So I'm assuming it's either the switch or a broken wire

 

A broken wire would be no horn. Check a service manual but I don’t think there’s a relay, probably a button wore out, unless it has an aftermarket configuration.

 

Welcome to the forum.
I don't have schematics for your bike but I believe the horn switch gets power from the Accessory fuse on a Orange/white wire. From the switch the Yellow/Black wire goes to the horn and from the horn to grd is black.

If you mean you have continuity going through the switch then you know what the problem is. If not pull the connector going to the left hand switch pack and check those 2 wires to confirm the switch is stuck.

 

Was it OK when you first got the bike and if so have you done any mods or work to it? A common problem is trapping wires when refitting the handlebar switches, so you may simply have a trapped and damage wire. The horn button could be stuck or defective.

 

I'm not sure the horn wires were disconnected when I bought bike. The previous owner did install chrome housings and buttons so something may have been damaged during installation