Okay...I made some progress and then found that my unit is not assembled the same as Jeff Erickson's in the Tech Tip.

I got as far as getting the needle off and starting disassembly of the odometer - removed the shims and found unexpected components. Got a little further and I am now stumped.
After removing the shims, there was not enough play to shift the odometer shaft the way it is stated in the Tech Tip, "Now the odometer can be removed by sliding that side in, releasing the geared end which can be lifted up and slipped out."
At this point, at the opposite end of the shaft from the gear, there was a brass spring clip which I figured out how to remove...but now keeping the shaft from moving far enough to free up the geared end, there is a brass sleeve on the shaft. There are two "grooves?" on the shaft which may tell someone how the sleeve was installed, or how it should be removed, but I am hesitant to just blunder ahead as I know that I would rather live with an old looking odometer that works, rather than one that was destroyed during disassembly.
I have taken photos to try to show the unexpected items.
Any suggestions about what I am looking at and how to proceed would be very welcome!
(Still wondering what grease and oil options to go with during reassembly.)