If the flywheel turns when you move the key to the start position then the starter is engaging and the solenoid is doing its job. It sounds like you have multiple problems. First, check the battery voltage. 12.6 volts at 80 degrees is a fully charged battery. You may need to charge the battery or replace it if it won't hold a charge. Second, your engine running issue after it starts sounds fuel related. Give us some more information, please. Approximately how many hours are on the engine? Please list model, type and code numbers. A 2011 model may still be under warranty so check you bill of sale and owner's manual for warranty in-service date and warranty period.
Ethanol in gasoline has been creating havoc for small engines, which have not been engineered to run on ethanol fuel. If these engines were engineered to run on ethanol fuel we would not be experiencing the problems so prevalent since ethanol was added to fuel. Manufacturer's will not warranty ethanol related problems so your problem may not be covered but it is worth a phone call to the dealer you bought it from.
Tackle one problem at a time and start with checking battery voltage.