I have a hunch that my problem stems from the varied sizes of the layers. I went into Illustrator and made each layer the same by connecting the path ends outside of the artboard so that each shape had the same width and height. For example:

This prevented the paths from jumping around the screen when I copied each path keyframe into one layer’s path. Progress!
However, the number of points varies so much from shape to shape, that things get wonky even when I set the first vertex or change the direction of the path in Illustrator and reload the footage.
Phooey.
UPDATE! I found a solution, but I’m not sure why it works. I put a rectangle in each of the Illustrator layers, gave each a stroke, and reloaded the footage in AE. When I created vector layers from the new files, they contained two groups, one for the rectangle around the frame and a second for the path I want to animate.
For some reason, when I copied the paths as keyframes into one layer, they stayed put. Well, they bumped a little to the right, but nudging the position was simple. All I had to do was take the stoke out of the first group and everything looked great.
Could anyone explain/point me to a tutorial on why this worked?