I had a similar issue; however mine is a '03 with a 5.9.
So keeping that in mind, when mine was in the "crank but not starting mode", I was not getting voltage from the coil when cranking. While this issue was intermittent, I changed a bunch of parts; however, this issue devolved from being intermittent to being a permanent no start condition.
(Which is easier to troubleshoot.)
Keeping that in mind, & shortening the rest of my story, the rest of my troubleshooting ended in not getting voltage at the trigger wire (when cranking) from the PCM to the coil (which on mine is a black wire w/ white stripe), as evidenced by back probing trigger wire with a test light & not observing it flashing (when cranking).
I realize that your issue is resolving itself almost immediately, so that makes it harder to troubleshoot.
However, if your issue devolves as mine did, if you get no voltage from your trigger wire to coil, the troubleshooting tree I followed drove me next to the ASD relay (which I had already swapped) & then to the PCM.
I am going to plug SIA Electronics in (I believe) Tilden Il., simply because they solved my problem. I contacted them & sent them my PCM. I cannot remember what the bill was, but I remember it being under $150.00. They told me they replaced the "coil transistor" & "resoldered a loose pin". They do have a fee for checking & finding no defects & shipping back, but I cannot remember how much it is, but it is probably on their web site.
So I'm just throwing that out there for you in case that's what your issue turns out to be, as they solved mine.
I tend to doubt your problem is your coil. My admittedly limited experience has been that coils either work or they don't. But I would think that you could get a coil from RockAuto.com pretty affordably if you want to do that as a check, & even if it doesn't work, at least you will always have a spare coil. (That's the rational I am holding on to for my extra relays, cam pos sensor, & crank pos sensor I now own).
I hope you post the resolution to your issue for the future reference to us all.
Good luck!