Is the spark plug fouled? Is the compression low? Bad rings could be a contributor. In which case rebuilding the carb is putting new wine in old wineskins.
Well their is only 4 ways fuel can get from the tank into the sump, in order of the ocurance .
1) overflowing carb
2) fairies
3 ) leaking pump diaphragm
4) young child "helping daddy"
Replace the carburetor fuel solinoid on the bottom of the float bowl or fit a fuel shut off valve in the fuel line, turn off the gas when it's not running. Also make sure the needle and seat is working properly.
In addition to the other tips already offered, if the key is left on when the engine isn't running, the carb solenoid valve will be open until the battery goes dead. When the fuel tank is located above the level of the carburetor, a leaky needle and seat can cause fuel to overflow from the carb into the cylinder and the crankcase. If this happens, the oil level will be high due to the fuel in the oil, and the oil on the dip stick will smell of gasoline. Diluted oil viscosity will be very low and can cause the engine to be ruined if run hard enough, long enough. I had a neighbor throw a rod on the engine of her nice mower because of this, and another neighbor left his ignition on more than a couple of times, but luckily it was caught it before the engine was damaged. An unexpected dead battery could be a sign that this happened. If there is a significant amount of fuel in the oil, it and the filter, if so equipped, should be drained and replaced with new oil before running the engine.
This didn't seem to post when i added above videos: Did you ever figure this out? I have the same problem on my Honda GXV620 engine. I turn off fuel and it will run, very rough, but runs. When fuel is one it runs, but sounds deep. Blowing smoke bad. I have put on an OEM carb and replaced the fuel solenoid and it's still happening. I have to change the oil after almost every couple hours because it's just full of gas and I don't want to blow the engine. What should I try next? Rings? Gasket leaking somewhere?