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.