[LMP]

First I suggest you edit your duplicate post and delete it..so all replies are in the same thread.
Not sure if 3.1 reacts similarly but with the 3.8, a faulty crank sensor 3x signal cuts the injectors but will still procure spark using the 18x signal..so I'd think investigating the crank sensor..located in the 3.1 distributor housing. Crank sensors are heat sensitive when they age and what you describe is typical . They run OK cold, and cut once warmed up....