1) it is rather improbable that a variable inductor is failing, as long as there was not a huge mechanical shock.
2) there were two different circuit layout variants. In both, the total inductance of the tank circuit is about 68uH. This is obtained in the older variant with a variable inductor of 68uH and in the newer variants (much more often) with a fixed 22uH inductor in series with a variable 47uH inductor.
3) Systematic diagnosing of the volume circuit consists of performing these steps systematically. That means do not go to the next step while the previous does is not accomplished.
- checking the volume oscillator without the volume loop connected and with an oscilloscope probe connected directly to the tank circuit. You should find there a sine wave of about 500kHz, 22Vpp, centered around 12V DC. If not, check all oscillator's components and eventually replace them until you get this. You'll need a "true" oscilloscope for that, a simple software solution which uses the sound card input of your computer will not work since the frequency is too high.
- checking the linearization coils for DC continuity: The 2.5mH coils should show a DC resistance of about 8 ohms, the 5mH about 15 ohms.
- setting up the theremin on a mic stand (since operating it on a workbench risks to mute the volume loop) and checking the DC level at the anode of D1 (take care to keep all cables away from the volume loop!) while changing the oscillator's frequency by tuning the variable inductor through the whole range and back. There must be a peak during that where the DC voltage changes by about 4V. If not, replace D1, even if it seems ok when you make a simple diode test with it.
- as soon as you are on this 4V peak in the tuning curve, let the variable inductor in that position and check with your hand near the volume loop if you can make voltage change back by approaching your hand to the loop. Either the volume circuit is working now or there is still a problem with the VCA pre-processor (1st half of the LM13600), but this would be a very rare phenomenon.
Don't hesitate to report your proceeding and progress back here so that I can help you in a more detailed way if you can't accomplish one of the above steps.