The best advice here would be to install a fuel shutoff valve, and turn it off and let the carb run dry. I don't know if he said it had a fuel pump, but if it does, that is one possible leak. A vacuum checker can determine if the diaphragm is tight. The other source, is the needle is not fully seating, and flooding the carb and then leaking through the engine to the oil reservoir. and/or filling the muffler.
You can change the needle, but if the seat has corrosion on it, then the needle still won't seat. To remedy that, requires a small brass brush, small enough to reach into the cavity for the needle and rotate the brush around a bit. Using your finger to push on the float is not good enough to check the needle and seat. A lot of pressure will surely shut off the fuel. The best way to check the needle and seat is to let that small amount of fuel in the bowl to close off the fuel flow.
Do yourself a favor, and check the oil level before starting. If the fuel level in the tank is lower than it should be, find out where the fuel went. It either leaked on the ground, into the crankcase, or someone stole some fuel.
If you happen to get water in the engine from where ever, the water will settle to the bottom of the sump, and if you catch it before you crank the engine, you can prevent it from emulsifying and you have a mess. Never did figure out how the water got in the crankcase on some of my engines.
If your gas tank should empty into the crankcase, you might get what is called hydraulic lock, whereby the engine will start to turn and just lock up. Remedy is to just drain the oil, refill both, and use the fuel shut off on the engine, and let the engine run until it stops from lack of fuel. If you want to know how I know all this, it is that I have experienced these things, from 22 years of landscaping, using Kohler, Honda, and Briggs engines. Shut off the fuel and cover the engines is the best advice I can give.