Got a bit further with the etch a sketch at the last WIP night at Foulab.  There are at least two components to the backlash problem. One - a curable one - was that the way I was using the AccelStepper and AFMotor libraries was not ideal.  I was releasing the motor immediately after finishing the stepping.  This caused the last step to be left incomplete, at least some of the time.  Introducing a 500ms delay before releasing the motor cleared this up.

This was detectable by writing a very simple program that took just a few steps (in this case 3) very slowly.  Then I could count the steps by eye, and see that some were missed.

That cleared up a lot of the irregular wobbling.  However, there is real backlash in the etch a sketch mechanism. When the direction of movement is changed, there are nearly 10 steps where there is basically no movement, and another 5 or so where the movement is proportionally reduced.   It would appear that this is due to stretch inside the etch a sketch itself, not in the gearing.  It persists even if the other axis is moved in between the forward and reverse movements.

You can see this in the picture - each step in the zigzag is ostensibly the same number of steps.  The squished ends are due to this backlash.

Just to be clear, i moved the cursor down and left to the bottom corner of the drawing shown, then shook the sketch.  Then the motors were moved in a zigzag to the right, then the reverse pattern.  The squishing at the ends is due to the backlash - note that the backlash in each direction is almost perfectly equal, meaning that the ends line up.  Gives hope for correcting it.