cold start and quiet problem

[Santy]

Hi, my dad's 2004 250 6.0 has a cold start problem, the weather here in Puerto Rico during the nights it is not very cold. But every morning the engine idles roughly, and when we accelerate, the cabin shakes a little. The other problem is that when it is in idle, the engine just stops making the loud sound they do, it quiets down, and when we give the accelerator a tap, it goes back to normal, but the it quiets down again. But know it is doing it when accelerating from a stop. I think it could be a faulty in injector, the fuel control module, or the high pressure pump. I need help before this problem damage the entire engine.