Your problem, or at least one of them is U1B. You have both inputs grounded to disable it but that is wrong. It will act like a comparator with both inputs the same value, so internally it can be toggling current in the output stage enough to show up as internal noise for the entire IC.
To correct the problem and disable an unused op-amp the correct way tie the (+) input to ground (or ref voltage) and the (-) input to the output pin. Now it is a buffer with zero or ref volts input and a closed feedback loop.
Problem 2 is that you have a single-ended 12 volt supply, so it is best if R3 has a 10uF capacitor in series with ground, so the DAC becomes the only DC reference. The capacitor will still allow AC signals to be amplified based on your gain settings. However if it is working with the offsets you have and giving you the correct DC response, leave the circuit as it is.
Below are a few schematics to clarify grounding of un-used op-amps.
simulate this circuit – Schematic created using CircuitLab