[ImperialCrown]

Welcome to Allpar. The fuel pump should 'hum' right after key-on for a second, then turn off waiting for the engine to crank over. The delay doesn't sound normal and may be a relay, wiring or PCM problem?
Get a wiring diagram and use 12 volt bulbs with jumpers to the fuel pump power/ground terminals and work forwards to the underhood fuse/relay (PDC) box. With long enough wires and a bulb tucked underneath a wiper blade, you can watch the bulb when the engine quits to see if it goes out or not. A fuel pressure gauge screwed into the rail and tucked under a wiper blade can help monitor fuel pressure as well. This will help locate the problem.
The P0352 isn't necessarily the coil and is probably a symptom more than a cause. It can be wiring/connectors, PCM or just where the engine decided to stall at. Even with 2 plugs not firing, a P0352 should at least give you a sputtering 2 cylinder engine with the other half of the coil working.
The crank position sensor can fail and not set a fault code, but I'm not saying that is the problem. Diagnose first. Does your scan tool display sensor and input/output states from the PCM?

 

[Muncle]

Do you remember if the old fuel pump had a delay like the new one? It might be the how the aftermarket pump works.
I had a similar problem with my 98 ram turned out the camshaft sensor was broke, the "ears" where it bolted down broke somehow and it was moving around causing it to act strange and sometimes not start or sputter. I dont know about the breeze but on my ram the camshaft is located under the cap and rotor. I just got one out of a junkyard and replaced it, worked fine. I would replace the cap and rotor and check or replace the camshaft position sensor.then maybe a crankshaft sensor.
Does anyone here think an bad egr would do it??

 

[thebluesguy54]

I don,t know what engine you have but on my dodge stratus 2.4 fuel injected I had the similar problem with stalling and it struggled at higher speeds on the freeway. I suspected it was fuel pressure,or carbon in throttle body, or perhaps fuel injectors , or coil, or Idle air control motor(valve some call it) also timing and plugs and wires were all suspects. Plugged cat was also on the list along with a vacuum leak. It was none of these! Well after 4 weeks of messing with this engine I finally pulled the egr valve off,New just last year,and mind you i had no check engine lights for my problem. I pulled the transducer valve apart and found it had melted the diaphragm. Also i noticed the line going to it ,which is exhaust gasses, and carbon had almost tottally plugged it. I did replace my IAC for the second time in a month..the first one was not functioning at all, got it from advanced auto. this car now is running beautiful after replacing egr.

 

[Bob Lincoln]

Melted EGR is often a sign of a catcon plugging up.