1)This is not a cooling system issue, however...
a)You say the cooling fan runs constantly and will not shut off?
b)This could be a relay issue, a controller issue or the vehicle is in "limp in mode", which seems unlikely since you say there are no fault codes stored.
c)Is the "power loss lamp" illuminated while the engine is running?
If the power loss lamp is not illuminated, you need to check and verify basics.
2)Warm climate rules out condensation freezing in the Map Sensor line.
3)You will need to connect a fuel pressure gauge, verify static fuel pressure is correct, start the car and watch fuel pressure as the engine runs and when the car shuts down.
a)Did fuel pressure drop as the engine ran?
b)Was fuel pressure low when the car shut off?
*The wiring may be different at the pump and the ASD Relay will be different year to year(84,85-87,88-90,91-up) but the test is basically the same. *MPFI (multi-port fuel injection) works on pressure differential across the injectors. *Since the injectors are exposed to manifold vacuum total...
www.turbododge.com
c)You can rent a fuel pressure gauge at AutoZone for free, you leave a deposit which is fully refunded when you return the tool.
If fuel pressure is OK, continue...
4)Verify the ignition system is capable of producing enough voltage.
a)Use an "adjustable spark tester" to verify the system is capable of 30 KV.
b)If the system is capable of maintaining 30 KV, start the car and allow it to run until it shuts off. As soon as the car shuts down (do not turn the key off then back on) check to see if you have spark, hopefully the car does not start right back up.
c)When the car shuts down do you have spark?
If the ignition system is fine, continue...
5)With the engine running perform a wiggle test on the wiring at the HEP, logic module, power module and other connectors/harnesses underhood in an attempt to duplicate the problem.
6)Be sure all engine grounds are present and securely attached to a clean surface.
a)The main controller ground on your car comes from the injector harness connector and bolts to the fuel rail, there should also be a ground strap on the R/S (passenger) engine mount bracket.
7)Be sure to closely examine the logic module connectors for signs of corrosion/oxidation or other damage.
It was a common problem as water tends to leak into the R/S kick panel from the antenna mounting.
a)If corrosion is present clean it using a soft toothbrush, Q-Tips, baking soda and water, dry with compressed air and then use an electrical contact cleaner or 91% rubbing alcohol to clean any oxidation from the terminals.
Post back with your results or if you have any questions, concerns or run into problems.