Schematic review - Darlington transistor seems to stay saturated in rare cases
I think the problem with this circuit is that 1) zenerdiodes are leaky when the reverse voltage is below their zener voltage and 2) the darlington Q5 only needs very little base current to turn on.

The leakage currents of those zener diodes might be enough to turn on Q5 when it is expected to be off.

Maybe you can solve this by adding a resistor between base and emitter of Q5. I'd start with a 100k resistor. Then the leakage from the zeners has to be large enough to make enough voltage across this resistor.

You're also right about the lack of a (current limiting) (base) resistor in the path BE of Q5 - D5 - D3 - D4. Since the base current of Q5 should be small (when operated properly, I mean, why use a Darlington if not for the small base current ??) there should be no problem adding a base series resistor to Q5.