No need to do any flushing , just fill with the premium blend.-----The issue is with the forward spring and clutch.----Stop running it and initiate repairs.----Easy fix in my opinion.----I developed a method to repair this about 45 years ago.
Tested this solenoid yourself ?-----Why would you need a new solenoid for the choke ?----What is your motor doing / not doing??----Does motor start right up with manual choke ?
???---Test the choke solenoid !!----Simple jumper wire from battery side of start solenoid the choke solenoid should make it click.----I doubt that solenoid is faulty !