When you cleaned the carb, did you shake the float? If there is a liquid sound in it, it needs to be replaced. The brass floats go bad, the plastic ones not so much.
Make sure the intake gasket is not torn. I think there might be an adjustment for the idle fuel mixture on top of the carb. The high speed is set from the factory.
Good idea to install a second shut-off valve.
The "sensor" is a fuel shut off solenoid. When you turn the key to 'on', the wire should have power to it and you should hear a clicking sound coming from the solenoid.
Surging is an indication of lack of gas. Did you check the float setting? You might need to blow out every hole in the carb again. Make sure the solenoid plunger is moving up and down when you turn the key on.
The governor adjustment would be my last choice to mess with. Very simple on Briggs and Stratton tho. Basically, you want to hold the throttle butterfly open at full throttle. Loosen the governor arm. Turn the governor shaft(goes into the engine) counter-clockwise while holding the butterfly open all the way. Tighten governor arm. I think I picked the right rotation. They very rarely move unless someone messes with them.