Ram 1500-engines stays running w/out key

I hope to get a response here.

Does it run like it normally would or is it normal then gets really ROUGH and tumble, sputtering till it finally dies?
Before I can offer advice, I would need to know the answer to this question to have a clue as to what the problem may be.

Your dealer misdiagnosed the problem then. This is called "run on" more common in OLD vehicles, but can be a problem in newer vehicles.
What has happened is that a layer of carbon has built up in the cyl heads, and is causing a symptom that used to be known as diesieling. All it is is that the carbon gets HOT, hot enough to ignite the fuel/air mixture after the ignition system has been turned off. If your amp is wired wrong, and the vehicle has fuel injection, this COULD be making the symptopms worse, because it could be providing power to the fuel system after the key is shut off. Ususally that system has no power, starving the engine of fuel so run on can't happen.

Run some very high octane fuel (even so far as to add an octane boster) for a tank or two to help burn the carbon off. You may want to think about switching to at least mid-grade fuel if not high octane. Their are fuel additives out there that claim to help burn off carbon deposits, you can feel free to try a couple different ones. I had success with only two of them personally zmax and lucas fuel treatments (2 fill ups apart). The only other cure is to have the heads removed, and cleaned, which I hope it doesn't come down to.
Get that amp wired correctly, FAT power should be directly on the battery, and remote should be to your power antenna lead on the radio. Fat ground should be to a solid body ground close to the amp. If all is correct, ask mechanic to properly diagnose based on this info here.

Another thing. I just thought of the "knock sensor" on the side of the engine. Have that checked or replaced (should be fairly inexpensive). What it does is detects "detonation" which is the combustion happening before the spark, and it will adjust the timing to correct for it and does a pretty good job at burning off the carbon o its own.
Any hemi will feel the effects of this more so than most other engines, because of the higher compression ratio. If you are using elcheapo gas it's possible that even normal operating temperatures are causing this run on(and not carbon build up), as cheap gas doesn't have to be compressed as far as high octane does in order to spontaniously combust.

Sorry for spelling and grammer, just had surgery and the pain meds make me not 100% here.