A failed capacitor doesn't have to be bloated like the pix in the link. Can look great and just not work. The pic following is a RUN Capacitor. It is actually two capacitors in one, serving both Compressor and Fan. It'll be labeled with two values in MicroFarads, such as 40 MFD and 5 MFD for Compressor and Fan respectively
You can tell it serves both because it has a total of three terminals. One is Compressor (may be marked HERM), the other is Fan, third is common to the other two.
The plastic one is a Start Capacitor, serving only the Compressor and should look about like this.
Some meters will measure Capacitance and give a pretty good idea if a capacitor is good or bad. Many techs will simply substitute a new or known good one.
Does this A/C use a Wall Thermostat, or does it have just a couple control knobs? I ask because I'm not familiar with the control boxes that go with wall thermostats. A bad part in the control box could be doing nothing but make a hum, not even trying to run the A/C itself.
If it's knob type, like ours, I'd suspect that dual capacitor had failed in a way that killed both Compressor AND Fan. What are the two MFD (might be expressed as uF) values, and the AC Voltage, marked on it?
Another test you might try, is find a clamp-on Ammeter. Amprobe is the name brand.
Measure the current draw by clamping over a power wire to Fan, then to Compressor, when you are trying to start the system. Best to have somebody on top with the meter, helper to turn system On and Off. Don't leave it straining to start (showing AMPS) very long. If the Capacitor is bad, AMPS will exceed the spec on the labels of Fan and Compressor. Sure, they could both be seized, but you can eliminate that for Fan if it spins.
Just for me, if an A/C has refrigerant leaks or bad compressor, it's junk. If it needs Controls, Fan Motor, Capacitors, etc. then it's repairable.
If God's Your Co-Pilot Move Over, jd
2003 Jayco Escapade 31A on 2002 Ford E450 V10 4R100 218" WB