on a 2010 Chevy HHR fan continues to run after engine is shut off

Some vehicles will allow the cooling fan to run for a short while after the engine is shut off. This may be normal. I cannot find any specific documentation regarding your HHR (also, it is helpful if you share which engine your vehicle is equipped with). Testing with a diagnostic tool that monitors component operation can see whether the fan is commanded "on" or "off" or what duty cycle it is commanded to, so that you can see if it corresponds with what is actually happening. If it differs, you could have a fan motor or relay issue. There are procedures to follow for testing.