High side Mosfet switch leakage issue?
My first impression was, that R1 and R2 are way oversized. 10M may be considered as an open circuit in some applications. 

Try lowering those resistances and see if your oscillations persist. 

If the oscillations stop you could raise the resistance again but bypass them with small value ceramic capacitors to provide a low impedance return for transients.

Designing low power systems is not easy and you will be tempted to use huge resistance values but they often cause problems like the one you describe.