Ok, so I've added another wire to my setup, from where I soldered the existing wire to GND trace (top-left end of the wire in the picture), to the GND pin on the Due board, the one next to Vin (I soldered the wire to the bottom side of the board). I've also fixed the trace I've previously broken on purpose. I now measure thermistor-ground-to-due-ground voltage no more than 0.002V, and temperature measurements have also improved. I think turning bed on and off only offsets the ADC by 1/1024 (still haven't gotten around to using it in 12-bit mode, but this issue suggests it may not be worth it...).
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