I went back and meticulously re-read your first post, and attempted to the best of my abilities to re-do everything that you suggested- just as you suggested, including the steps to familiarize myself with the behavior of the meshing mechanism. Everything behaved just as you described.
I re-performed the meshing procedure following your instructions and those in the documentation for the mount. That procedure still did not ellimnate the play in the RA axis, as it DID when I performed the same procedure on the Dec axis. Because that worked, I conclude that I followed the procedure correctly.
I then added your last step to move the lever to the 90 degree position from fully locked, and gently with one finger pushed back and forth on the end of the counterweight shaft - and it still moved as it does when the lever is pointing to the fully engaged and locked position.
After all this I re-checked the worm nut - with the cap removed, I cannot feel any movement of the worm shaft when moving the counterweight shaft with my finger.