The ohms reading problem could very well be a contact problem, especially if measuring 2 wire ohms. If 4 wire Ohms, it depends on how the cables are connected.
For the mV readings, there could be some thermal EMF from holding the DMM in the hand. The 3478 is a bench top meter, not a hand held. I would not expect that much trouble, but for best stability it should sit on a bench / board and not moved around.
The first points to check would be the supply and the line frequency setting (50/60 Hz). Also check the integration time - the preferred setting for manual reading is something like 10 PLC or more. Another point would than be looking at the DC zero reading in the 30 mV / 300 mV and 3 V range. This should tell if the problem is more with the ADC or the input amplifier.