And while we are speaking of engine shut offs - there are often two of them. The normal engine stop lever on the top of the governor, activated by an air cylinder controlled by a Skinner valve, and the emergency stop flap valve on top of the air intake manifold, activated by a electric solenoid and the emergency stop switches. That one is known to randomly apply itself and have to be manually reset, that happened to me twice. Sadly, it took just as long to figure out the second time as the first... Symptom is the engine cranks and lots of white smoke comes out.
Reading your original post again, engine starting and running fine and slowly dying as the air builds up is classic engine stop lever problem. What happens is the air supply to the Skinner valve that controls the engine stop air cylinder is fed from the accessory system, which is after the Pressure Protection valve. The engine must run long enough to build up 65 psi of pressure before the PPV opens, then must run long enough to build up pressure in the accessory system to move the engine stop air cylinder and the effect is the engine stop lever is very slowly activated and the engine just slowly dies.
What smoke appears as you crank is a good clue. If lots of white smoke pours out the exhaust, that means there is fuel and no air. If no smoke at all comes out, that means there is no fuel. That can be either no fuel (duh!
), or the engine stop lever activated.