I had a similar issue, actually the same issue, with a Briggs engine 2 months ago. If the float is not working or the needle valve is stuck open fuel will continue to flow. And not only will it continue to flow, but because of the way most Briggs small carburetors are designed, with a vent to the crankcase, the fuel will fill the crankcase mixing with the oil.
The first clue will be very large clouds of smoke if you can get it started. A fuel shutoff is not the cure. Replace the carb and replace the oil. Keep checking the oil for about a month until you are sure that the problem is resolved.