..And it's not thermocouple effect either. If so, it wouldn't change that quickly with supply voltage.
..And the CMRR is typically better than that.
I have seen this before; It can happen when you connect an excessive capacitive load and/or antenna (meter lead) directly to the output of the amplifier.
Squinting at the video I can't make out the print on R9 (the output series resistor Dave likely put in the ucurrent for the same reason), it is populated, but the print looks a bit short to be '101', and it's not in the DaveCad*.
A bit of capacitance across the feedback resistor would probably be an idea, too. There should be 500mV p-p noise coming out of it without one.
Come back next week to see if our AD-caped hero defeats the evil Doctor Offset.
*Generating netlists in this is a pain.