Several issues have been pointed out that would not be fixed by replacing the wobbly worm gears.  Each issue could be addressed individually, but this would be time-consuming.

Therefore I've decided to reduce the scope of my efforts.  I plan to continue testing the Pi and DC motors, but to use a conventional, cartesian robot that will be similar (or identical?) to existing open-source designs.  This will reduce complexity and make debugging much simpler.

Since I'm rebuilding the robot anyway, I might as well invest in higher-accuracy sensors.  Perhaps I'll get some optical encoders.  Alternatively, I might make an attempt at using computer vision- others on this website have had success with this approach.  I would have to purchase and calibrate a camera of sufficient resolution, but OpenCV definitely has the capability to do this sort of thing (at least to moderate accuracy).

I'll probably post something to this website when (if) I get another robot built and working.  But it will probably be a few months (or even a few years).