The end of the saga has been fortunately reached! I thought there was a problem with the ohms circuitry, but I went into DCV mode and put 30V through a 10M resistor into the 7075. Since at that range, the input impedance is also 10M, I should get a reading of 15VDC. Which I did, but only after about 20 seconds of ramping down from 30V to 15V. So that led me to believe that the input impedance of the 7075 was varying, as though a capacitor was somehow in the circuit and charging or discharging.
And so it was. A 7406 controlling various relays had a stuck line which forced the ACV circuit to be permanently switched in, hanging right off the input lines regardless of whether I was in DCV or Ohms mode. Some input/output capacitors in that circuit, aren't there?
Anyway, a new 7406 fixed all ills.