Could be your valve seals leaking oil into the cylinder and upon start up it burns it off. But I guess first what color is the smoke. Black is fuel and blueish color is oil.
As far as your leaking overflow, when you cut off the fuel shut off that stops the flow to the carb so it would stop the leak, but the leaking overflow is from either floats not adjusted properly or they are sticking. Since you say it leaks after sitting awhile I would say they are sticking. You can just tap the float bowl with a wrench or hammer and it usually frees it up.
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-2007 Honda 400ex (Hmf slip on, jet kit, K&N)
-1999 Yamaha warrior (stock)
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