On my '98 12V, the relay (located forward/middle in the panel) is a spare, connected to nothing; it may have a function in your '99. One of the two circuit breakers (side-by-side, near the center) controls power to the power seats, which you don't have. The other controls power to the door locks and windows. The large combination relay (forward, bottom) controls turn signals and 4-way flashers.
If the radio stays on, I would first suspect the key switch, especially at its age. Remove the knee bolster (should be three screws at the bottom, then 'clips' at the top) and locate the connecter to the key switch and figure out how it disconnects. The next time it happens, disconnect the key switch. If it stops, you may have a faulty key switch. It sounds like the switch is not completely disconnecting the power and enough current is getting through to power the radio and almost power the relay.
It takes thought and patience, but I believe you can remove the key switch without removing the steering wheel. If so, disassemble it if possible; a good cleaning may solve the problem.