Perhaps I didn't make things clear enough in my first post. If the radio and the cigarette lighter share a common hot lead, and that hot lead is connected properly to both, and the radio has a ground, and the cigarette lighter doesn't work, I think what is happening is that the lack or failure of the cigarette lighter's ground is causing it to seek ground through the radio. I suggest that as a first step, you replace or add a dedicated ground to the cigarette lighter and see if this solves the problem.
If one circuit which shares a hot line loses its ground, it will follow the common connection through whatever other component has a ground. This kind of problem is pretty typical of a failed ground.
edit: I should add that one characteristic of this failure is that the wrong connection is drawing its current through the other component, which is not a good thing, and probably explains why the head unit is shutting off.
