I remove the key and the engine still runs, what is wrong?

What is your question?

I have a 2011 Ford Fusion with remote start. Twice now I turned the key off and removed it but the engine still runs. With the key back in I wiggled the wheel and it turned off, the second time I put my foot on the brake pedal and it turned off. I am replacing the battery on the key in case it is the remote start. I have been told several possible causes; everything from reset the PCM, to change the PCM relay to replace the ignition switch. I would like to start cheap and work my way up.

What makes this problem better or worse?

its the same and usually late at night

How long has your 2011 Ford Fusion had this problem?

a week