I have a 2006 F 350. It takes some time to gain oil pressure when cold then it may run 10 minutes or may run a minute before it throws a code for the cam timing and drops oil pressure. The idle drops and the cams get all out of phase.
I have replaced the left bank cam phase solenoid because it was leaking oil from the electrical connector but that hasn't helped. If I rev the engine my oil pressure is fine and there is not much valve noise. Any ideas where to look next?

 

It is possible you may need to update the engine with an increased capacity oil pump .Could be looking at new chains, solenoids phasers etc.

 

We bought the truck in this condition, I didn't even hear it run until it was towed to our place. I thought maybe the pressure relief valve might be sticking but I am inclined to search the valve train.