Doing this from memory because I don't have my horn apart any longer. When you honk the horn it activates an electromagnet that pulls the diaphragm toward it. At the end of it's travel it makes contact with a ground that breaks the circuit and the diaphragm returns to it's starting point. This is done quite rapidly, resulting in the vibration that is the "honk".
The horns on my 38 depended on a stud protruding from the diaphragm to make contact with the housing, breaking the contact. The first time I put the single horn back together I put the diaphragm on 180 deg out & it would only cycle one way, basically what you describe. When I finally figured out what the problem was & turned the diaphragm around it worked OK.
The point of this is, you need to make sure the ground mechanism for the diaphragm is working OK & not insulated by rust or debris. To do this you'll have to disassemble the horn(s). Cleaning the normal electrical contacts may not be where your problem lies.
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