On 12 year old AC fan works, but there is a periodic muted buzz but compressor doesn't engage.

check your ground straps and bcm for connections-damage or failing

I would guess a loose electrical connection. Have you had work done lately?

Could be the magnetron or the transformer. No it is not worth repai

I passed this question the first time I read it, so I will take a stab at it now.
The fans don't come on until the compressor has pumped the freon pressure up to say 150 or 200 psi. And the compressor clutch has to have voltage to engage the clutch.
I wonder if the clutch is going and needs 14 volts instead of 12 to engage ? Or the relay that powers the clutch ?
Speeding up the engine would increase voltage from the alternator.
This is assuming the serpentine belt is tight and everything else is working correctly.
I think I would swap the relay with another and put a meter on that circuit to the ac clutch to see what it takes to engage it.

So what does the fuse that blows, protect?

You need a wiring diagram & start to test
the components most likely to cause a high
current draw & blown fuse.

Don't keep replacing it.

You know the fan control circuit works, but the
compressor doesn't stay on. Did you monitor
the compressor as the fuse blew?

May have a module for compressor & fan
If so one side failed
The compressor clutch could be bad

Fan motor is probably bad, if it is under 5 years old the parts are under warranty. Do not run the unit for very long without the fan working as it can damage the compressor.

Check the fuses.

It is strange that the compressor engages when the button is on, but the light does not work.

Does cold air blow out the vents? If so, the light may have burned out...

It sounds like your evaporator fan motor, they last about 4-5 years. Try to see if the buzzing noise comes from the motor, usually in the freezer compartment behind a vented cover plate

The freezer is usually cooled by the compressor, the fridge is then cooled by an evaporator fan motor ($30). If this fan seizes or slows down then the fridge warms up (it sometimes starts squealing before this happens). It could be that your drain hose is clogged or frozen. Maybe the temperature adjuster go bumped.

If you think it is the evaporator fan motor:
Check to be sure you can get a replacement fan first, they last about 4 - 5 years. Can get from a parts supplier or electric motor company.
UNPLUG fridge. Located usually in top of freezer behind a vented cover plate. Remove cover, remove screws holding fan in, may have to cut wires. Replace with new motor, make sure the fan pushes air into the freezer area and doesn't spin backward drawing air from the freezer area.

Are you fully charged though? If you don't have a full charge the clutch will not kick in.

You have one of two problems going on. Your fan is spinning at hitting the sides. Or your compressor is shot which seems to be more the problem and even though it is cold the humming sound is a warning for you that the compressor is shot and it is more affordable to replace the fridge then the compressor. Sometimes compressor can cost more then the fridge.