I've been in this situation in the past. Amplifying microvolt-level signals with good SNR is a nightmare.
I can spot two mistakes in your circuit.
The first mistake: R2 and R5 are way, way too big. The thermal noise voltage of a (combined) 44k Ohm resistor over a 20kHz spectrum at 25°C is 3.8µV RMS, which is already on the same order of magnitude as low-volume audio signals from a dynamic microphone. The noise from these resistors alone will be louder than some of your audio signals. You can calculate the noise power density of a resistor using the formula for Johnson-Nyquist (thermal) noise.
The noise contributions in your system are (referred to the input):
- Microphone (150 Ohms): 1.6 nV / sqrt(Hz)
- AD620AN: 8 nV / sqrt(Hz)
- Low-pass resistors (44k Ohms): 27 nV / sqrt(Hz)
Multiply this with the square root of the bandwidth and you get the RMS noise voltage. As you can see, the bigger the resistor, the bigger the voltage noise. To fix this, you can remove those 22k resistors entirely (replace them with wire links) and place the capacitor of your RC lowpass directly across the mic's output. Then you're just using the mic's well-defined internal resistance (150 Ohms) as the RC filter's R. Alternatively, you can also use 100 Ohm resistors without significantly degrading the noise performance.
One way or another, the 22k resistors have to go, though.
The second mistake is in your rail splitter: The AD620's REF input is not high impedance. In fact, it's got an input impedance of about 20k Ohms. As a result, your virtual ground (generated by those two 18k Ohm resistors) is swinging all over the place and causing the AD620 to oscillate and malfunction in all kinds of exciting ways. You have to use an OpAmp to buffer the virtual ground.
Additionally, the AD620AN itself has a rather large input equivalent noise voltage (it's about an order of magnitude noisier than the mic). Consider using a lower noise instrumentation amp, i.e. the AD8429, which is an order of magnitude less noisy than the AD620AN.