Why are you making a simple square-wave inverter? it is good for driving incandescent light bulbs or an electric fry pan.
It will poorly drive an electronic speed control on an electric motor and will not drive many electronic products that rely on the higher peak voltage of a sine-wave.
A pure sine-wave inverter does not use a linear amplifier, instead it uses a complicated pulse-width-modulation circuit that switches the Mosfets on and off at a high frequency. The lower frequency sine-wave is produced by varying the on and off times. The Mosfets switch fully on and fully off so they do not get too hot.