Try replacing your fuel shutoff solenoid. This is an electronic device in the injector pump that is responsible for cutting fuel when you turn the key off. Otherwise, the engine would never shut off. If this malfunctions it can cause issues like what you describe. It can also fail in the opposite position, making it impossible to shut the engine off short of cutting the fuel lines. You may consider calling a local diesel shop and ask them if that is something they have in stock and if you can replace it with simple tools. I have never touched a cummins, but I know they use Bosch pumps and Bosch pumps on VW use these solenoids.
Next thing to check... there are small steel fuel lines connecting each injector to each other. I think those seal with a copper washer gasket. Make sure those are tight and not leaking. If air is introduced into the system it can cause the engine to stall or run rough. Make sure all other fuel lines are in great shape and tighten down all the clamps. Good luck.
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