Ok, considering the switch is new...
Headlights and interior lights have a direct connection to the battery, hot at all times, bypassing the ign switch.
If it was only a starting issue, I would guess transaxle position switch, but since the accessory position doesn't do anything either...
Have you checked for voltage at the ignition switch?
If you don't have the dash apart already, you can do a little testing at the power/ground distribution box.
These are fusible links, the one I am holding in the picture is for the ignition switch. It has an orange/rust colored wire (the fusible link), and then it splits into two solid red wires. Hopefully you have a multimeter, test for voltage on the red wires (one probe on ground/black, the other on either red wire). There is one the splits into solid red and red/white, that's not for the ignition switch. This is all located right by the battery tray, the black cover hides most of this...
If the red wires test ok, test for voltage at the ign switch.
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Brando