Get a meter.They are like $10.
You can`t work on cars w/o one.
You don`t have a short.You have the opposite-a high resistance connection.Heat is generated there.
" continuity across the ignition wire." What does that mean? 0 ohms?,1 ohm?,10 ohms?
A poor connection at the connector is the source of your overheated wire.If you put a voltmeter across it,there will be a voltage drop under load.
When you relpace the switch,the connection is still bad
I recently replaced my ignition switch because of high contact resistance.I eliminated the connector at the same time by soldering the wires.
ELIMINATE the CONNECTOR
The ignition switch and all wiring needs to have very low resistance because both the solenoid and ignition coil have load resistances on the order of 1 ohm.
An additional 1 ohm across the switch and connections means they see only 6 volts under load-not good.
I measured up to 6 ohms across the ignition terminal of the switch depending on how you wiggled it.
Ran quite well except for a cold misfire(surprising),but it is way better after switch replacement-silky smooth now.
I had to make an ea81 tilt switch from a normal one by cutting off the tabs and soldering wires on ,but,that is another story.
One wire IS hot all the time.
If there is still nothing on the solenoid after repair,check the neutral safety or clutch switches.