There is a diode under the dash which is used for the buzzer, it ties into the ignition relay. If this diode failed(shorted out), it would send power from the room fuse(10A) back into the ignition relay, then into the meter fuse(10A). From there it would go to the ignition switch`s output. Which could keep the car running until it blew a fuse.
If this is what happened, then the fuse ROOM would probably blow, followed by the METER fuse.
You said you have no check engine light with the switch in run, so the first thing I would check are these two fuses under the dash.
Edit: If one of them fuses are blown, unplug the diode or remove the ignition relay no 1 before replacing em