What I would do is to hold a probe on one center socket and turn the coarse switch back down and stop when you see a resistance reading. Let us know what that resistance is and what step you're on with the control.
I am somewhat confident that the meter you have just isn't properly reading large resistance values, though it is advertised as being able to do so. If you lose the DC resistance value on both center pins of both 6C45P sockets at exactly the same step on the coarse switch, I would tend to blame the meter. If the resistance reading drops out on one side but continues counting up on the other and then drops out, I would suspect flaky solder joints on the coarse switch itself (though doing a ton of resoldering on these switches is not at all recommended).
If the meter is the problem, the ~$20 meter at Harbor Freight can resolve these measurements.