Kevin,
Aren't your "104" capacitors actually 0.1uF? At 60Hz, a 0.1uF cap has a capacitive reactance of 26.7K ohms. Therefore the 10M resistor across one doesn't change the phase, nor does the 500 ohm resistor across the other.
Your circuit is nothing more than a single-stage integrator circuit with a DC gain of 20,000. Single-stage integrator circuits don't oscillate. Maybe your "oscillator" is just amplifying the ripple voltage of its power supply.
The opamp's DC gain of 20,000 is interesting. If the opamp has an input offset voltage of 6mV, its limit, then the DC output of the opamp will try to be 120V! So the opamp's ouput will be saturated against a supply voltage, even if you used a "typical" opamp with an input offset voltage of only 2mV.
Your new transistor amplifier stage is also interesting. I wonder why you added it since if your opamp circuit did oscillate, it doesn't have anything to limit its output so it certainly wouldn't need an amplifier.
Without any negative feedback, the transistor's output will be very distorted. Also, since the current gain of transistors vary about 8 times from one batch to another, you can't predict the DC voltage of its output without having any negative feedback:
The 10K bias resistor has 11.3V across it providing 1.13mA base current.
1) A transistor having its minimum current gain of 38 will have a DC output voltage of +9.85V. It is almost cutoff.
2) A transistor having its maximum current gain of 300 will have a DC output voltage of -4.95V. It is almost saturated.