Thanks for the replies
I have now let the lambda pro plugged in for a couple of hours, which helped a lot about the crackling. Then I opened the frames and discovered gaffa tape around the connectors which touched the stator. There is a resistor soldered to the connectors on each side, too. There wasn't really much dirt, I just removed the gaffa tape.
There is still light crackling and squeaking but that may be related to my head and ears moving and touching the inside of the cups.