I don't know much about the XJW-01 design, but the 12V DC PSU you mention will probably have a Y capacitor for EMI suppression between the AC and DC sides which could introduce noise, apart from general pickup by the metalwork and cables. Maybe something else was switched on in the house/building that was injecting noise into the mains. Using a battery as the power source could help isolate the issue.
Having one hard earth connection somewhere is generally best to avoid ground currents. Alternatively, if the XJW-01 has a guard terminal, you could connect that to the metalwork instead of the earth. Ideally the guard is via the coax shield around each force & sense connection.
If you were to use a bench LCR meter with an earthed chassis, then that would possibly provide an earth connection as part of the guard (shielding) in 3 or 5 wire measurement configurations (2 or 4 wire plus guard). Depending on the design and use case, I think the guard will either be driven by the test signal, or bonded to the force low (which in turn may be earthed). Apart from any coupling affecting measurements, there is also a risk in having the force signals floating and potentially being a hazard when injecting bias voltage or currents from external equipment.
This would normally be covered in the manual supplied with precision equipment.
Some discussion about guarding for low level measurements are under this thread. Mike Wyatt who started that thread would know a lot more about this topic than me.
https://www.eevblog.com/forum/testgear/measuring-small-capacitance-with-ks34465a-th2830/You might find additional relevant info in this amazing doc, although it doesn't specifically related to LCR or AC measurements
https://download.tek.com/document/LowLevelHandbook_7Ed.pdf