Mike,
I've encountered similar symptoms. The cure is shown in this link, just at the beginning:
installing a remote ford solenoid/ chevy starter | Grumpys Performance Garage
After installing the remote starter solenoid my click-click problems went away. It seemed redundant to me at first - why would I need a solenoid in series with the one on the starter? All I know is that it works on both my big block cars. I also think Ford would have done away with it instantly since it cost more - and yet they included this setup for many years.
If I have a click-click now, I know the battery is going, so I replace it and all is well for another xx years.
If your car already has this setup, perhaps the solenoid on the firewall is bad, or maybe it's the battery.
Hope this is some help,
Tom