EEC-IV (OBD-I) will not be much help for this kind of problem.
Their might be a remote switch and speaker test.
It's hard for me to remember that far back.
All test are valid.
Where is the audio module located. While I'm trying to remember 2 decades back.
Is it a separate module that could be mounted in the drivers door, trunk or mounted to the floor. Any and all these location could have a moisture problem.
From post #1
I got the radio to work but it won't stay working after cycling the ignition.
This indicates that their is a power loss at times. This can be a power loss internal or external of the radio or a loss of ground.
After checking fuses, corrosion on the connector side of the fuse box and all ground connection including the ones behind the kick panels where many main ground are located for the main dash harness I would check the ign switch and if their are remote switches check them.
***A little trick.***
If you have a friend that can help.
Turn the radio on and do a wiggle test and have a friend listen for any changes in the static or normal sound.
Ford audio system are very sensitive; it doesn't take much to make them go goofy.
Let me know how things go with the antenna or whatever.
If you have a friend in the auto junk yard business you could do the swap parts test on main components. Doing this could save a lot of time and money spent.
In the Dealership we did this to save time but we used our parts department.
The bad news. You could have more than one problem.
If you have any question just ask and I will try to remember back and come up with something. I did this kind of of repair for 40 years and much of that time in Ford Dealerships. All I have to do is remember.