The solenoid on the carb bowl is a anti-backfire switch. When its working you will hear it click when you turn the key on, and again when you turn the key off. Since you don't here it now its either stuck, clogged, the wire has come loose, or broke. They are always going bad. It's there so when the key is turned off when the engine is running full throttle, it has less chance to back fire when the engine stops.
When they go bad, I just pull it off, extend the plunger and cut it off flush with wire cutters, screw it back on and tie the wire off out of the way.
That might be your problem cause if its stuck in the UP position, its cutting off fuel to the engine.
If the engine is flooding, either the float has gas in it, or the needle is stuck, or the needle seat has expanded and the needle can not shut off the flow of gas to the bowl. If it's a gravity feed system ( no pump ) it will allow fuel to fill the crankcase, and if you run the engine it will wash the cylinder(s). Thats bad.
A inline fuel filter AND shut off valve should be used. Always pull the dipstick and take a whiff, if you smell gas, drain and fill with fresh oil after the flooding problem has been fixed. Don't run it if you smell gas in the oil!
Hope this helps
Rob