Ether Jones wrote
The only way I can figure that gasoline could get from the tank to the
exhaust is through the float valve in the carb. Do most inexpensive
carbureted riding mowers nowadays have a float valve in the carb and no
shutoff valve in the gas line??

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Yes.

You didn't ask, but a common cause for your problem is a
small bit of debris holding the float valve open. The cure is
a simple cleaning. The prevention is to be extremely careful
to get gasoline and only gasoline into the fuel tank when
refueling. It doesn't take much of a bit of grass clipping,
dust, dirt, etc. to cause a problem.