I have a B/S model 303777 it will only run on full choke line from tank is clear and fuel pump is working. Could it be the fuel solenoid on bottom of carb and is there a way to check it thank you

Hi,
Yes you can by removing it, plugging it up and grouding the body of the solenoid. Then turn it on and the center should retract if it is properly grounded and working. They can gum up and be cleaned with spray carb cleaner and working the plunger in and out with repeated sprayings.
Something about the carb is dirty or not working. The solenoids are very reliable, but do gum up.
Peace,
Clarke