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!