My experience with the factory horn wiring is not as described above.
The horn relay is a three terminal single pole, single throw relay. The horn button on the stalk is a momentary switch on the ground side of the circuit.
+12v is supplied to one relay terminal via a short red jumper (from an adjacent relay) to the coil and armature (the moving part) inside the relay body. Pressing the horn button provides a ground to the relay coil terminal via the brown wire which causes the relay to throw. The armature makes contact with the third terminal which sends +12v via the white wire to the horns. Release the horn button, no more ground to coil, relay disengages.
The horns are actually the only electrical accessory up front that are grounded to the body, as opposed to the fans, markers and headlights, that all use one skinny little black wire all the way back to the stud behind the dash.
Somehow your brown wire is finding an unintended ground, or the relay is stuck.