The current noise floor of a resistor sitting on a table using unshielded cables is something like 1 nA. If you want the maximum digits of a 1G resistor, all of the cabling, connections and the device must be shielded.
If you want the offset of the cable leakage to be subtracted, you have to guard the cables and the device. A guard enclosure inside of a shielded enclosure will eliminate noise and leakage offset.
It is not surprising that a 7 1/2 digit DMM measuring a 1G resistor appears to be noisy. Its probably not the DMM's fault.