73' Hard top shuts off unless you hold the shifter in park.

Hi guys, Im new to the forum and I recently bought a 73' Mustang hard top with a 302 and C4 transmission. I've recently replaced the fuel pump, water pump, and starter solenoid. The guy I got it from had it torn apart but I figured I could work on it. So here is my issue--well, two issues. 1. When I turn the key, the starter fires up and the engine cranks and runs, however, the starter will not shut off. When I back the key off a little bit the whole engine shuts off all together. I was able to disconnect the starter from the starter solenoid and with the key in the "on" position I can crank the engine by touching the starter wire to the starter solenoid and when it cranks I disconnect it.. I was able to have the engine idle for an extended period of time while I tuned the carb. This brings me to my next problem. If I am not pushing on the shifter all the way into park the whole car just shuts off like a kill switch.
I purchased a new starter switch and am going to troubleshoot the wires tomorrow, and from what I have researched.. the transmission problem could be linked to the neutral safety switch? Am I going in the right direction? any other pointers? also, the ignition switch seems like a pain to get to. Any info would be greatly appreciated. Thank you.
Tommy
 
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There are 2 small wires on the starter solenoid, one is to jump 12v to the coil during crank, the other is the start wire that puts 12v to the s terminal to crank the starter, they may be switched. Does the engine start to crank as soon as you put the key to the run position? Take both of the small wires off the solenoid, turn the key on, then use a jumper to put 12v to the "S" terminal to start the engine. When it starts remove the jumper. It should stay running.
 
I've done that and the engine stays running. I was able to tune the carb while having it set up like this.. The car only starts when I switch to start.. when I back it off a bit to kill the starter the whole thing shuts off
Hi sorry.. I can post a picture of the setup on the starter solenoid... if you'd like.. I appreciate the advice so far..
 

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OK, I may have misunderstood. Do you have a test light? If you don't buy one, they are cheap. With the key in the run position you just need voltage at the "+" wire on the coil. That power comes from the ignition switch, threw a resistor wire, (I think It's Pink). And if I recall there is a bullet connector plug at the firewall with the ignition wire and the sender wires in it. If you don't have power at the coil trace it back until you find it. If you need a little hands on while your doing it I'll give you my test # if you PM me. I'm generally putting around my shop on the weekends.
 
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