Well I took it apart to clean and inspect the inside. This thing is
built to last, not a piece of plastic to be seen! (Well okay, the
strain relief for the power cord is plastic.) I figured out where the
slight rotation is coming from, and it doesn't look like anything is
worn out or loose. The shaft slides along a block and the top of the
shaft has a channel in which the pin from the motor slides to create
the saw motion. I think the slight play actually is designed to
minimizes the impact on the motor. The thing runs true and doesn't
wobble around when it cuts.

In my infinite ignorance I managed to back out the blade holding screw
too far too forcefully and stripped the thread near the tip. It still
seems to hold a blade but I was pretty mad at myself. If only I could
find a cheap one for spare parts...