Why do you think you're hearing "priming" instead of just "pumping"?
Here's how it works, because there is a safety issue with having the fuel pump continuing to run with the engine off, after say an accident, the PCM shuts the pump off after @15secs.
So, if you roll the key to the On position, you're only going to hear the pump run for that long. That's the way it's supposed
to work... It's not running so as to build pressure and satisfy a sw, it's just timed-out
The "Shutting Off" at idle is certainly a problem though, unlikely to be fuel starvation though. You'll have those issues at high rpm first because the pump can't keep up. It's really not using much of the pumps output to idle. I'd be thinking something else...
Does your truck have a Trans Temp Gauge in the cluster? (There was a Major electrical revision during the '01 run) If it does not
have that gauge then you can see the function of the CPS (a more likely culprit) in the Tach. If, while you're trying to restart after it stalls, you see the tach needle move up slightly (indicating engine rpm on the starter) the CPS is functioning. If you do
have the gauge, maybe just change it! EVERYBODY should carry a spare in the glovebox anyway, so even if it didn't fix this you'd have your old one as a spare.
You can always test if you've actually starved for fuel (maybe you have another common issue in the tank with a cracked pick-up) by NOT letting the pump run after it stalls and opening the drain (yellow lever behind the filter housing) and see what comes out (3/8s metal tube against the motor, above the pass ft tire) and comparing to what's there after you let the pump run, key On, for a cycle or two. That will tell you if the bowl was empty when it stopped running.
How's it run otherwise?
Is the tank below 1/4?