Chris wrote:

I'm finding I get different results (vastly different in some cases) when
measuring the total resistance of a circuit with a) a DMM and b) an old
analog meter with a physical needle. And this doesn't only happen at high
impedance points, either. What could account for this?

I've got four DMMs and two analogs. The DMMs agree with the other DMMs
and the analogs agree with each other. But the different types don't
agree with each other!
I have some older DMMs. I get odd resistance readings the first time I set
it to Ohms scale. By wiggling the plugs on the probes at the meter end, and
occasionally working the range selector dial around a few times, I get a
stable Ohms reading of about 0.4 Ohms with the probes shorted. Then, I get
more sensible readings on circuits. So, these meters get poor contact on
the range selector switch and the bannana jacks. So, that is one thing to
check for.

Second, most DMMs use very low voltage to meke Ohms measurements. Analog
meters often used 9 or even 22 V batteries for the Ohms range, to push
enough current to move the needle on high resistance circuits.

If there are any seminconductors in your circuit, a DMM likely will not give
enough voltage to forward bias any junctions, while an analog meter will.

Jon