readings on VDC are steady. I'll try to setup something to measure currents and see how it looks. Also I ordered some better test cables, but will take a while to arrive.
That means, that the reference and the DCV path is fine.
DCI mode will not change anything, as it does not add any active components .. you can skip that.
Focus on Ohm mode, as this adds several active circuit elements to the DCV functionality.
If you can really exclude external noise sources, the problem will be buried inside the Ohm circuitry ONLY, it's no use to search somewhere else!
For 100k and higher Ohm ranges, the test cables should be short like 10cm or so, and they should have good surface metallization, like gold plating, at least it should not be oxydized.
I'm not in a nosy industrial environment, I'm at home, I have a laptop on the next desk like 1m away. It does have a power brick under the desk.
"nosy .. environment" - that's great , if you live in a whisky distillery

(sorry for that joke..)
Sometimes, you can't imagine where switch mode PSUs are located.. I had a new LED lamp, and the mains adapter produced a lot of disturbance, so I had to throw that lamp out of my lab again.. So look for further possible sources .. or simply change the location of the instrument.
Maybe you repeat this experiment w/o the laptop PSU plugged in..
Trying now with a 10k ordinary resistor from the parts drawer. I'm reading 9.870,22 Kohm and the last 2 digits fluctuates like this 22 24 26 22 20. so this is MUCH better than on 1Mohm range. Also the reading slowly changes. now last 2 digits read 41 42 43 42 43 43 43 44 45 44 45 46. The unit was cold and there are less than 5min since I turned it on, so it probably warms up. this was done in 2W mode 6 digits FAST. also the resistor might warm up a little and this migh show since it is not precision/high stability.
The 34401A needs half an hour at least to stabilize.. please don't make any observations before that time, that makes no sense.
The resistors self heating can be neglected! That's about 100µW, you will hardly see any effect, if you use thin film resistors (50ppm/K), and wait a minute for stabilization, at least.
The 10k range should be rock solid, (using appropriate cables), that means, the last of 6 digits (fast or slow) should only change by +/- 1 count.
What happens if you toggle the front/rear switch, several times?
Does that change the reading afterwards?
(Some units suffer from a defect / dirty switch, especially in 2W mode).
Frank