Ok. Doesn't sound great, but is common to both the units we have.
The AD797 is configured with a feedback capacitor of 1.5 nF parrallel to to a 3.08 kΩ. This is for a 34 kHz pole. Could it be an issue with the choise of capacitor type? I'll try switching out the one that is in there for another mica one, I would have thought the one that is there would be enough (unless it has bad HF impedance).
I'll try some things.
If you have an oscillation in a circuit, it is no longer linear!
Sounds like a typical AD797 oscillation. GBWs of OP27 and AD797 are 8MHz and 110MHz, respectively. OP27 can't make such high freq oscillation.
I didn't check the circuit diagram. Usually AD797 has ~5pF feedback capacitance to increase the stability. You may be able to stop the oscillation by adding/replacing a mica capacitor from 10~470pF. But 50% of the time the oscillation is not fixed with this kind of modification. You need a cut-and-try.