Since it was an emergency, I didn't have the time for anything more than to fire off the original NNTP request. It turns out, I had to learn on the job, due to the nature of the emergency.
However, *now* I have time to learn where I erred.
Googling, I find this nice YouTube video that shows how to clean out a shower drain using the same Cobra drum snake that I have (mine is branded Brasscraft because Home Depot asked for their own differentiating brand).
I had always thought that you run the motor continuously; and that the reverse was for rewinding the cable; but both those assumptions are wrong.
Apparently the forward position is only used *at* an obstruction; and the reverse is only used *momentarily* to get out of a bind.
Of course, they don't tell you how to rewind all that 75 foot of cable *back* into the drum. I called Cobra today and they said you push the cable forward and rewind it always without the motor running.
The only time you use the motor is when you can't push or pull anymore. Then you only use the motor momentarily they said.
So, in this case, even the professional YouTube videos are lacking in the detail needed to actually perform the job properly. You only know enough to ask the questions if you've already done the job itself.