The "right" (lowside) MOSFET gate should not have 150vpp on it. I would guess it should have 20-30Vpp on it at most, centered on the negative power supply rail. Scope out your negative supply rail as well - probably there is a lot of ripple on it causing the source of the MOSFET to be swinging around wildly.
The "left" (highside) MOSFET is more difficult to measure since the MOSFET rides on the output node driving the output transformer, which itself may well be 150vpp at the switching frequency under normal operation. You really need a differential probe to measure Vgs of the highside MOSFET since neither gate or source are DC with respect to ground.
That said, apart from a possibly badly decoupled power supply I think it might be working normally. When you drive audio in you should see jitter on the waveforms. If you see nothing but a stable switching then probably the fault is in the input stage that pulse width modulates the switching signals