Sounds like a diode has gone in the alternator which is allowing current from the battery to backfeed through the blue indicator circuit wire from the alternator to the dash clocks, which is then holding the ignition circuit live.

An easy way to test this (there are others but I find this definitive) is to disconnect the alternator warning light bulb and see if the engine stops. If it does, alternator is shot, if not, then you have other issues, which means disconnecting things until it does stop (which could cause other problems).