AFAIK there is no available schematics for this DMM. So one would likely need to do some kind of reverse engineering / guessing from the signal what function the chips have. So start with a good photo and maybe text information from the manual / service manual if available.
Sometimes one could follow the signal path by applying a low frequency test signal to the input and check a few amplifier outputs. Usually the non AZ mode would be easier for this.
Another way to look at it would be to start with a few more readings. So check some +- 1 , 2, 4, 8, 10 V in the 10 V range and a few tests in the 1 / 100 / 1000 V ranges. Another point to check may be the input current: so what is it reading with a 1 or 10 M resistor to ground. Sometimes the exact error symptoms can narrow down the fault. One would than still need to locate the parts on the board.
The DG444 is not super low leakage, so I don't think this would be directly switching the input signal. The DG444 may be used to set the amplifier gain or maybe controlling JFET gates. I could also be used to switch an already amplified /buffered signal.
The waveform may indicate something like auto zero switching (between the input and zero) - one could check this by comparing AZ on/off mode. Without a plan CR36 does not tell us anything what signal is measured. With the DG444 or OPs one could at least tell if the signal is an output, control signal or supply.
The voltage looks somewhat odd, as there is no fixed level. So this could be something like a FET gate signal or similar, or a consequence of a broken part. The signal definitely noise not look healthy. With the higher voltages this looks like oscillation of maybe an open input (this can happen with JFET switches if the gate drive does not follow).