Hi,
Wow, you have really gone to great lengths to figure this out, and I wish I had an easy answer for you. I am not sure what you are asking with regards to your question about an audio balanced input. The AD8137 was designed specifically to drive ADC inputs like the ones on the AD1938. The Vocm pin allows you to move the reference for the AD8137 to align with the CM voltage, so the you can remove the AC coupling between the op amp and the ADC. Your original conception looked very good to me, and you should be able to remove the AC coupling from your circuit. I am trying to think of new issues to consider or other tests that you can run to narrow the problem. Do you have any standard single op amps that are unity gain stable? Maybe you could try to insert one into this circuit, with only a few modifications to the wiring, in order to test the rest of your setup. I am wondering if there is something else wrong with your test rig.
Regards,
Coleman