Howdy,
Fuel stabilizer is not really much of a factor in less than 1 year old fuel. [BUT, having said that, drain the tank and put some fresh fuel in it to eliminate the fuel being a problem. You can always use it (diluted with fresh fuel) later]
Your problem is more likely an adjustment problem. When you pull the throttle back to the idle detent, it should NOT be placing excessive "CLOSE" {vs open] 'pressure' on the throttle butterfly control.
You determine this by removing the cable connection AT the carburetor ...........and doing the idle adjustment [in NEUTRAL] prior to re-installing the throttle cable (this allows ONLY the carburetor return spring to close the throttle butterfly)
THEN, you adjust the throttle cable at the carburetor so that it can be reconnected without placing any additional "close" pressure on the carb butterfly linkage............
Regards,
Rick