My courtesy light fuse keeps blowing

A simple trick to finding a short is to replace the blown fuse with a pair of spade terminals, then use small jumper wires to connect them to a 12 volt light bulb. A brake light bulb works well. When the circuit is live and the short is present, the bulb will be full brightness and hot so be sure it's not laying on the carpet or against a plastic door panel. Now you can unplug electrical connectors and move things around to see what makes the short go away. When it does, the bulb will get dim or go out.

For intermittent problems like yours the bulb may be dim already. Watch what takes place when it gets bright. That's when the short is occurring. It could be due to the rocking of the engine when you shift between reverse and drive. It could be due to the body flexing when you drive over bumps in the road. The bulb limits current to a safe value when the short occurs, in this case about one amp. That's enough for the radio, but the circuit needs more current when the interior lights are on, so everything will likely stop working.

The radio has two fuses. The one for the station presets and clock memory isn't labeled for the radio. That circuit is always tied in with some other circuit that is always live. That could be the horn, the brake lights, the cigarette lighter, or most commonly, the interior lights. My guess is you have courtesy lights in the doors. If you do, broken and frayed wires between the door hinges is a real good suspect. The test bulb will likely flicker when you move the doors.

A rapidly-clicking relay is one common symptom of a run-down battery, or in this case, low voltage due to that test bulb. Try a larger bulb, like a head light bulb. That will allow up to five amps to pass, still a safe value.