Why does DMM resistance measurement not settle when probing pins on circuit?
I've seen this happen before with my Agilent (and other) DMMs. When you measure a resistor directly, the meter is essentially running a small current through the resistor and watching for the voltage across the leads.

If there are other components involved (like the ones on your assembled PCB) you can get some odd measurements back - particularly when there are capacitors or transistors in circuit that store or switch the energy added to the circuit by your meter.

It might be helpful to turn the auto-ranging function of the meter off.

One of my favorite features on your meter that I often use for finding shorts on a PCB is the four-wire resistance mode. If you can scrounge up four leads you can put the meter in this mode and it can measure the resistance with greater precision. This probably won't help your "jumping around" issue very much, but the extra precision may help guide you as you work to get closer to the short(s) on your board.