I had same issue with my horn. Turns out I had the little metal disc on wrong. The piece I'm talking about is the metal disc with the holes for the three phillips head screws to go into. I had it concave, but it needs to be bulge toward driver so when you push the horn it makes that disc flex to ground out to make the horn sound. Maybe that could be your issue? Mine would sound if I even slightly pulled the signal arm toward myself and it would sound the horn and sometimes it would stay on. The good thing is you can disconnect the horn wire at the horn and you can still listen for the relay to know if the horn would be going off without aggravating yourself with the constant horn going off.
My turn signal switch was nasty inside and a new one fixed a lot of my rear lighting issues. This is the one I ordered, they also carry one for a tilt column if yours is a tilt one:
Non-Tilt:
https://www.classicparts.com/1967-72...tinfo/53-509A/
With Tilt:
https://www.classicparts.com/1969-72...tinfo/53-510A/