Cascaded op amp instability, oscillations
The  circuitry has a problem because you are getting your feedback from the driver output, which introduces a phase shift; when added to the phase shift of the two stages, your negative feedback becomes positive.  The driver is only capable of driving current into your output capacitors; when U1-B output goes high, Q1 quickly charges C1 and C2 (and overshoots a little in the oscillating case).  However, when U1-B drives in the other direction to compensate, Q1 does not drive the capacitors but can only turn off, and R7 must discharge the caps (hence the phase shift).  U1-B goes into negative saturation before this occurs, and then must recover into its active region.   When it does it overshoots again.  The first stage does not add anything to this circuit but does slow the response and adds to the phase shift.