Welcome to JUST ANSWER small engines, let me try to help you !!
# XXXXX is fuel shutoff solenoid. Remove it and see if it is sticking. Hook up the wire to it and with it grounded, see if it retracts each time the key is turned on. It can be restricting the flow of gas to the engine. It requires full battery voltage to open and stay open.
This will loosen the float bowl, so clean the float and needle valve and bowl while you have it off.
Here is link to the carburetor.
Mine does not have a solenoid on the bottom of carb
I had the coil checked out, they said it was weak so I'll try it tomorrow. If it does not help I will be in touch. Thanks