Someone, somewhere, has installed a relay, and miswired it, or miswired the voltage regulator voltage feed circuit, and it's running on the alternator.
I can't help you much beyond that, as far as by text help...
However, this you might want to check...
Is there a load on the battery with everything off and the motor not running?
Check by pulling the battery cable and hooking it back up.
If it keeps running after turning the key off, somehow the ignition is energized. So... One simply has to trace why or how it has power. Obviously, not the ignition switch. If someone has miswired a relay, it would most likely power an electric fuel pump, and it acts like a latching relay... it powers itself, once it's on.
Or, someone has "hotwired" the rig to get it to start, and the ignition is actually "always" on. That should not be too hard to figure out.
Also, there is a relay that energizes the ignition, bypassing the ballast resistor while you crank the engine.
This relay is rectangular, usually not far from the battery, and has a heavy wire between it and the battery. if it were stuck, or the wire were bypassed to battery voltage, it would not shut off.
This is all I can think of right now.
