Sounds like someone might've added a clock, perhaps the person who installed the junkyard struts, and didn't get the wiring quite right.
There are four wires to the clock.
A constant 12V+ for memory when the car is off, which sounds like it's a problem in your car.
A ground wire.
A switched 12V+ which provides power only when the key is turned to a powered-up position, which will make the clock display the time.
And a wire from the positive side of the gauge light circuit, which dims the clock slightly when the headlights are turned on. This wire also sounds like a problem.
My '93 didn't have a clock, so I snagged one at the junkyard along with its electrical connector and the four pigtails.
I wired it into my radio wiring, and it worked mint the first time.
I think I figgered out the radio wiring by searching online for a pin-out diagram of 1990s Toyota radios, and then finding the appropriate wires coming from the back of the radio. I probably used my DVOM to verify the wires, but I don't remember because it was well over five years ago.
Maybe I found a pin-out for the clock? Maybe the clock wiring are the same colors as the radio wiring?