I decided to pull the plastic choke housing off the carb just to inspect it.
I compared it to one from another one of my Holley carbs. The one I pulled off seemed to have a much weaker spring. I put the other, stiffer-spring, housing on the carb and the choke now works as I expected.
The original one was set two or three notches rich, but I have had to set the replacement one at the center mark as the fast idle didn't come off fro 5 minutes or so...!
It looks like it was just a bad choke spring. Or maybe the guys who rebuilt the carb just reused the old one because it looked to be in good shape.
Either way...thanks everyone for your input.