The manual does in fact say that if the results are unexpected, to try reversing the leads.
That sure seems like an indication that semiconductor junctions are involved in the measurements, so I think you will need to use an old-school multimeter.
The gradual increase in readings is very likely due to capacitance being charged up in the circuit under test. If my theory holds water, that capacitance would shunted by a semiconductor junction if the multimeter was causing that junction to conduct, and then the charge-up effect would not be noticeable.
So what VTVM did you get? The key thing, in this context, is that it needs to apply more than about 0.7V to its test leads when measuring resistance. You could check that with the Fluke! Just connect the Fluke (set to measure voltage) to the VTVM (set to measure resistance).