My tractor mower starts after sitting for a day or so but shuts off within about 30 seconds and does not start again. I can crank it up but it seems that the carburetor gets overflowed and fuel leaks out the air filter compartment. It is a B&S twin engine tractor.

Perhaps The Needle And Float Are Sticking Causing The Flooding, Check Your Engine Oil To See If It Is Overfilled, It May Be Due To Excessive Flooding While The Engine Is Off. If Overfilled Change Oil And Filter Immediately.
Remove Carb Float Bowl, Spray Carb Cleaner Up Center Emulsion Jet Tube(where Bolt Or Solenoid Screws In Through Bowl) , And Spray Around In The Bowl And Bottom Side Of Carb. Try This And See If It Helps.
If Your Carb Has A Fuel Solenoid On The Bottom Of The Carb. (cylinder Shaped With Either One Or Two Wires Attached), The Solenoid May Need Lubricating And Working Of The Actuator Plunger To Free It Up , Or Your Wiring To The Solenoid May Have A Short.
May Need To Carry This To A Shop.