Obviously, you have current back-feeding the coil.
With the key off, a test light @ the coil will prove this. Then start going back along the Ignition wire to find out what has shorted into the circuit.
Start with checking the coil for voltage with the key off. then since you just replaced the switch. go to the plug and unplug it and see if the voltage went away.
If so, then it's either the electrical part of the switch or something it feeds.
When the switch is in run, it supplies current to the coil and other things. with it off, that current source is cut.
But if somewhere in the circuit, something it feeds on the Run circuit, like a relay is not
turning off with the switch and has current going back into that branch
of wires to feed the coil.
Just a thought, unplug the windshield wiper motor and see if that solves it.
Ray in Mena