The fuel pump will run without a relay on these trucks. The pump gets voltage from the oil pressure switch, in case the relay fails. Later models don't do this, but yours does.... as long as there is oil pressure, the pump will run, even if you shut the key off, which is why you sometimes hear the pump run for a brief moment after you shut off the key.
If it's flooding, however, that's something different. The fuel pump doesn't have the ability to flood the engine, that fuel still has to enter the engine somehow, either by leakage, or by the computer misfueling the engine. The fuel pressure regulator is inside the intake manifold, and they're known to leak. This usually results in a hard start when hot complaint, not so much when it's cold. Cold flooding is usually due to a faulty coolant temp sensor, which erroneously tells the computer that it's much colder outside than it really is, resulting in way too much fuel and an engine that won't start due to flooding.
A scan tool that reads live data would reveal the coolant temp sensor issue. When the truck is ice cold, the intake air temp and coolant temp sensors should read identical, and they should be equal to whatever the ambient temperature is. So, if it's about 70F when you go to look at it after the truck sits overnight, then the coolant and intake air temp should be about 70F, too. Replace any sensor that deviates more than a couple of degrees from that.