Yes, I had a 74 charger that had a button, but wasn't sure on the a-body.
The charger also had little orange box with no button also

Guess I don't understand how it can crank with the yellow ignition wire unhooked.

Something, a circuit, is shorted out
All things broken are fixable

I think I'd try replacing the relay first and eliminate the easy stuff.....
my 2 cents,

Joe

You said you think someone has been into the wiring. Can you show or tell us why you think this?

Can you send a picture of the wires going into the relay?

I am hoping the wiring has been messed with and the colors are off.

I am starting to think, like cab, that you have a wiring short but a picture of the relay and wiring may help.

Did you check to see if the reverse light stays on through all the gears?

Do you have a voltage meter? A regular test light?

look closely at the column connection where it plugs into the main harness.
this is a common place for "melty's" to occur.
please let us know what you eventually find.

Any resolution to this problem??

Joe

i'm wondering as well.

Update. Thanks for all the help. Someone had spliced a wire from the yellow ignition wire down to the solenoid wire by in addition to it going to the relay. Finally found it tucked into the wire harness. Seems fine after unhooking it.

glad to hear the problem has been solved, and what the cause of it was.