I've got a band-aid that seems to prevent the thermal runaway from happening but I'm hoping you can provide some insight into "why" and perhaps a better solution.
I found some nasty looking oscillations when scoping the gate of Q11 to see why it was turning on - several volts of unstable crud in the 600-800 kHz range. A 1nF cap to ground reduced the amplitude and shifted the frequency to the 225 kHz range, so I tried higher values. As much as 10uF reduces the oscillation without noticeable impact as now for the first time I have positive peaks and 100% modulation (even though I see some spiky things on the carrier which I suspect is the same oscillation).
The problem apparently is an instability in the modulator circuit that I'm just minimizing so the circuit can work but and hopefully this will enable you to get a handle on what's really going on. Meanwhile I think I'll try the radio on the air tomorrow morning!
73, Bob W9RAN