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.