If you're frying your contacts, you have too much current flow, not voltage. Running a 12 volt horn on 6 volts will not increase the current flow. Problems could be a bad ground the columnn to body. Also a bad connection, maybe the contacts in the horn. Also, check the wire size going to the horn and from the horn to the button. It needs to be either 10 or 12 gauge, I'm at work and can't recall which. Adding a relay into the circuit makes sense, I would assume that is how the harness you installed is set up, and using the wiring that is to power the relay (16 gauge?) for the horn could be the problem.
