most common cause is bushings in bracket behind steering wheel airbag are bad allowing contact at all times. However, I thought you stated earlier you could hear relay click on when pushed?
Regardless, test steering wheel bracket by connecting up good horn and either manually pull out on airbag assembly or jamb some cardboard or rags around the edges (stopping frame contact) for it to stop. If it does, issue is common bracket failure. Instead of having to dig in twice on physical horn, replace w/ your temp one (or another one) and just pull horn fuse until you can R&R bracket. For the bracket R&R, either replace entirely ($50-65 IIRC) or repair bushings (< $7 w/ Permatex blue silicone).
See this link for R&R links/more details.
If the horn doesn't stop when temporarily pulling on airbag assembly, you may have a short somewhere else but I suspect bracket is cause...
Good luck
