The engine stays running with the key off

Hello, This is likely a bad ignition switch or main relay. When you power the vehicle off, the ignition switch may not be turning the vehicle off like you requested. The main relay could be sticking on, and that would cause the same issue. Removing the battery cable is likely to reset the relay as you are removing all current flow and allowing the relay to cool. This issue will go beyond not shutting off, and risk battery drain, anti-theft system failure, etc... Best to have this diagnosed, but the technician will need the vehicle to act up in order to correct the issue. If you test the ignition switch (Electrical portion) while the vehicle will not shut off, you may find that the "run" position circuit is remaining powered.