Just an idea. My Father in law had a very similar problem. Seems like Sears changed the brass mesh fuel filter with a paper one. It just would not move enough fuel through the paper one. (with a half tank or less.) I would change the fuel filter with the brass mesh one. Maybe pull the carb and look into the fuel inlet. Blow with carb cleaner backwards up the seat. You may have a tiny piece of rubber hose blocking the flow. Full tank may have enough pressure to blow past. Until the fuel drops a bit.