Two-volts P-P looks like the correct P-P amplitude, but the waveform is unexpectedly different than the osc. square wave, and the distortion seen would not likely match at both the high and low output voltage conditions, should be most noticeable at the high output level, or more likely not be seen at all.
In this case, looks like a bad TL-082 op-amp or bad rail voltages.
Does the -4.4 volt rail look stable, meas. -4.4, clean, flat? If not, is C5 connected properly and the correct value?
[bPossible C9 wrong value or not connected across the meter.[/b]
Alternatively, one or more of the meter's bridge rectifier diodes (D5 - D8) are connected reverse or open circuited?
Put the scope leads directly across the meter. You should see almost perfect flat, but small DC voltage when DUT connections are shorted. Your scope should see about a plus 1 to 2volt DC pulse at the top of the 6.8k R21 connected pin2 of the op-amp.
You also could have the wrong values for C8, R20 or R21.
What is the waveform at IC3 pin 3?