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First, I am an electrical idiot.

Just for something to try, and eliminate the switch, try switch off. Lights on. Then pull on the 4 way flashers. Bypasses the ign switch.
This will probably still pulsate the clock.
I think there is a bad ground somewhere, you may have a bulb filament shorting across or something, or bad firewall connection?
The whole dash panel cluster surround is made of plastic on your car, make sure it is functionally grounded.
While the lights are flashing, walk around, looking for something that is on or pulsing that shouldn't be. Maybe something is being back fed. Dome lights, trailer wiring? If this is a Fla car and has an aftermarket hitch, it is almost guaranteed that scotch locks or that ilk was used to wire in the trailer plug. OK,ok, I've done it myself too.

If I am understanding what you are saying, the clock won't be lit up to see flashing because the key is off.

I have cleaned all the grounds and contacts. The thing I haven't cleaned is the positive cable at the starter, which I planned to do even though I can't see how that would be causing it, but I don't know. No trailer wiring ever. It is an old police car so it did have sirens and lights wired to front, so who knows with that stuff, but I don't see any obvious bare wires or wierdness. So short on time right now that I do a little at a time and then off to work.

Facts are stubborn things.