I'm not that familiar with the types of fuel connections one would find on this truck, but I've seen problems with multiple brands with problems like you're describing with old, original fuel lines in cold weather. Older rubber and plastics will seal when hot but can leak air past fittings and clamps when the temps go down.
Try this: Get a fuel can, fill with clean diesel. Run a hose to the pump's intake port. Start the engine and run it until you're sure all the air is out. Leave it hooked up overnight. Start it again in the morning while everything is at cold ambient temperature. If it runs correctly with no missing or stalling, you've got a problem with air leaking into your fuel supply lines. If it runs rough and wants to stall, then the fuel supply system is fine and the pump is in question.
I'm not sure if the p-pump is susceptible to this or not, but older Bosch mechanical rotary pumps (similar to the VE? pumps on later 5.9s) had a seal under the pumping chamber that would shrink over time, working correctly when ambient temps were high but losing prime and being a pain to start if left overnight in a cold environment. In the case of the P-pump, it's possible that the pump shaft seal could be pulling air in. Not entirely sure, but similar things have happened to similarly designed pumps.