Thanks, brother Indoz, you helped solve my problem. I put a meter on the solenoid posts, and no voltage is going to the starter while running. Yet, I still hear that clicking noise under the floor, near the starter. It's not an exhaust leak, but rather an odd thing.
THe original air intake duct from the grille that went to the old air cleaner was stuck down beside the radiator, and the alternator belt was barely clipping the end of the duct. I guess the sound was deflected back and down by the shaker assembly, and I couldn't easily see it from above. When I removed the negative/ground cable at the frame to clean the end, I must have dislodged that old duct.
The starter works fine, and all is well!!!
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BTW, the solenoid brand is Select, from O-Reilly's. I bought one with a lifetime warranty, 4 years ago, and get a new one each time for no charge. I have no clue why they fail so often, it was that way even before I bought the Bronco. I have new cables, too. I just be sure to always carry a spare one in my tool box.