I would like to add something, that I read. Even though you have a fuel shut off solenoid, there is fuel left in the bowl. If the fuel solenoid seat is not in perfect condition, it may allow some fuel to be sucked up the carb even after shut down. I believe the fuel shut off solenoid was developed for safety reasons, in case of a roll over, the seat safety switch would kill the engine and deactivate the fuel solenoid. A similar solenoid is used in many different engines to prevent dieseling. The Kohler on my Zero turn mower has a Kohler engine, with the fuel solenoid, and if some trash gets in the bowl, the engine would act up, surging and trying to shut down. After cleaning out the solenoid, it ran like it is supposed to.
There is a simple test that you could do to check and see if the solenoid will shut off the fuel, with out turning off the ignition switch. With the engine running at half throttle, locate the fuel solenoid wire, there may be two wires, one will be connected to the start terminal, and the other will be connected to the run terminal. and unplug or disconnect the wire from the fuel solenoid to the run terminal. and see if the engine will shut down. Check your electrical diagram to make sure that you know which wire is correct. The wiring for the Robin will be a little different. Jack Robin should be able to help there.
According to Kohler's web page, they say to shut down from half throttle.