Engine runs when cold but dies with no spark condition hot

I assume this is the front-wheel-drive model and not the rear-wheel-drive New Yorker / 5th Ave, based on the parts you mentioned. First you need to list the engine size. If you have the 2.2L turbo, suspect the Hall Effect pickup assembly in the distributor. Those were a high-failure item and a lot of people carried spares in the glove box.

With either engine current for the ignition coil, injectors, alternator field, and fuel pump or pump relay comes through the automatic shutdown (ASD) relay. To determine if that is turning on, measure the voltage at the coil or any injector. You will see 12 volts for one second after turning on the ignition switch. What's important is if that voltage comes back during cranking.

You have the front-wheel-drive model. Those had the 3.0L or 2.2L turbo. The rear-wheel-drive model is much larger and had the 318 c.I. V-8. They made those through 1989.

Once you take the voltage reading when the no-start occurs, if you find 0 volts during cranking, the most likely suspect is the crankshaft position sensor. There should be a diagnostic fault code related to that. It would have been erased if the battery ran dead but if the sensor failed again while the engine was running that code should have set again. Chrysler makes reading fault codes real easy. Cycle the ignition switch three times from "off" to "run" within five seconds, then count the flashes of the Check Engine light. There will be a series of flashes representing the first digit of the code, a short pause, then a series of flashes for the second digit, then a longer pause before the next code number flashes. Code 12 can be ignored and code 55 means "end of message".

Crankshaft position sensors often fail by becoming heat-sensitive and will work again after they cool down for a half hour. The pickup assembly in the distributor for the 3.0L causes very little trouble.

There's nothing special about the ASD relay. You'll find one in a salvage yard. They're a very low-failure item.

Wouldn't be a short on that wire because that would burn open a fuse link wire. Plus, jumping 12 volts to the coil would feed that short and your wire would burn up. You most likely have an open circuit meaning a break in that wire.

First of all, when you say the ASD relay is turning on, do you mean for that one second after turning on the ignition switch or during cranking? If it's turning on during cranking but there's no voltage at the positive terminal on the ignition coil, this is going to become a rather easy fix. You can turn the ignition switch off, remove the ASD relay, then jump the two wires with a stretched-out paper clip or piece of wire. The two wires to jump will be the two fatter ones, and one will be that green / black wire. Now you should have 12 volts at the ignition coil. If you do not, flex the harnesses to see which one makes that voltage come and go. Also look for a harness that travels along the left inner fender and under the battery tray. That one often moves back and forth during acceleration and when changing between drive and reverse.

Also check for 12 volts on the two smaller wires bolted to the back of the alternator. If you have it there but not at the coil, there is a corroded splice. If you don't have 12 volts anywhere, check right at the wires you jumped on the relay socket. If it's missing there, the problem is in the circuit that feeds the relay, not the green / black wire after the relay.