There should be a horn relay, under the starter solenoid at the right inner fender. It should have three connections directing downwards.
I think the most likely thing causing the horns to unintentionally go on, is the wire to the horn button.
The horn button grounds the coil in the relay to turn it on, i.e. the wire to the horn button is "hot", i.e. it has ALWAYS 6 volts on it, except for when the horn button is pressed (grounded).
If there is any damage to the horn button wire that can come in contact with the body , the horn will go on.
However, the "hot" horn button wire is connected via the coil to the battery, there is a resistance, and the current will be limited safely if there should be any short circuit.
The horn button wire goes into the steering tube, and because the wire will twist back and forth every time you turn the steering wheel, the wire i likely to give up after a while, and after 60 years it can be a little brittle.
So my conclusion go for the horn button wire first.
You can reroute a new horn button wire disconnecting the old one from the relay, which should be the far right one on the relay (but check first).
The far left connector should go to the horns, and the middle connector should be the power in. It is three connectors because both the horns and relay coil use the same power input.
If you disconnect the horn button wire on the relay, the horns should never loud. In that case the relay should be defective.
1955 Mercury