The situation would've gotten worse regardless.
The biggest mistake the crew made was to fly past three suitable airports while troubleshooting.
If you go back and re-read the transcript you'll find a short conversation about their concern for the overflights.
The jackscrew failure probably would've occurred sooner or later with or without troubleshooting. The very first concern should have been to get the airplane on the ground as soon as possible with any type of flight control problem.
I think the autopilot would've disconnected by itself when the final failure occurred because the stabilizer exceeded all limitations when the nut stripped off the jackscrew. The airplane was uncontrollable at that point.
I hope that clears it up.