Is this the correct procedure for checking motor shaft alignment?

And yeah, I feel you on the job description. I'm a machinist too, and half the calls I get end up being millwright work in a nice shirt. The upside is a machinist ussually has better feel for what a thousandth actually looks like than half the guys who do this for a living, so once you kill the soft foot you'll prolly be closer than you think.

Had something similar last summer on a pump rebuild. My readings kept drifting and it turned out the indicator bracket was sagging, not the machine. If your mounting arm's hanging out a few inches, gravity'll mess with you. I started zeroing the indicator, rotating back to zero after each sweep, and if the zero moved I knew it was the setup, not the alignment. Also worth checking if anything's tied into the motor, like conduit or piping pulling on the housing. I had a rigid conduit fighting me the whole time til I loosened it off. Once that was free the numbers finally settled down.

Spent a week chasing ghosts on a big horizontal pump once and it turned out the hold-down bolts were the whole problem. I was snugging them with a regular wrench and every pass I'd get a different sag on the feet. Finally borrowed a torque wrench from a buddy and went around in the same sequence every time, same number on each one, and the readings just stopped walking. Might be worth checking what sequence you're tightening in and whether the base is painted under the feet, paint squishes and acts like a spring. I hit mine with a flap disc to bare metal and shimmed from there.