Ha! Thanks! I didn't expect to be "recognized" here. :) I'm quite pleased with the response that Hidden in Plain Sight has gotten. It's a fun little game. :)
My problem appears to be solved. My first iteration was using the vertices returned by the texture to physics stuff to create loopshapes. That worked.
Then, I wanted to introduce lighting and shadows (using the Krypton 2D lighting engine... heaven forbid I should actually write my own code!) That required convex polygons, so I turned used the Bayezit decomposition to get a bunch of convex polygons,
which I added as individual static bodies. That resulted in the weirdness in the video. Seemed to happen mostly on long horizontal edges.
Then, rather than having a bunch of individual polygons, I created a CompoundPolygon instead, using the list of vertices. That appears to have solved the problem.
So... I can proceed forward with my work and there is no longer an issue. But I bring it up in case someone else is having the same sort of issue.