Check the key switch because solenoid should remain on (live) when key is in the run position (ignition on) and turn off when you turn the key to the off position.
I agree and that's what I'm thinking. The wire for the carburetor solenoid on the key switch is the same wire as for the starter solenoid which splits up in the engine compartment going two different location, to the starter solenoid and the carburetor solenoid and that's with the factory harness.
My key switch has 6 contacts marked with letters on them which I think stands for:
B = battery
A = accessory
M = magneto
S = solenoid
G = ground
R = not used don't know what goes there but when key is turned to on (not start) it gets power just like accessory