Need Help Diagnosing 01 V-6, Runs Very Rough

Trying to help out the daughter of a friend, and will only have one shot (this evening) to try and fix it, or at least figure out what is wrong.
It is a 2001, V-6. She said after a short drive, (like to a gas station), she shuts it off, then when she goes to leave, it is hard to start, and "shakes like crazy", and when she puts it in gear and pushes the gas, it does not move. Same whether in Drive or Reverse. If she shuts it off and waits about 30 minutes, it is fine. It has done this 5 times in past few weeks.
She said it does throw a CEL when it is doing this, so I may just drive it to the nearest Autozone, so they can pulled the code(s).
Any ideas, suggestions will be appreciated, only work I ever did on one of these was to replace a soft rubber PCV vacuum elbow that had rotted out. Thanks, -Jim.

The code will be important. The only similar I've seen is a transmission problem. The case that the clutches sit in cracks and no longer allows the fluid to stay inside that case. When aligned a certain way the clutches force the fluid out, no compression for movement, and the high pressure into the wrong areas of the transmission make it and the engine jump around out of balance. Again, the code will be the important first clue, so get that first.

Well, it started up fine, drove it about 3-4 miles to Autozone, code scanner revealed nothing. Drove it back, stopped & restarted it several times, no issues. Popped the hood, poked around, then went to park it, and it acted up. Not exactly as she described, but I would say classic symptoms of fuel pump failure. Would try to fire as it was cranking, and a couple times it did start, but running on maybe 2-3 cyls for a few seconds then died. Battery ran down very quickly, by appearance it is obviously old, & acted very weak. Thinking that since the FP is farthest away from the Batt, it "sees" less voltage due to length of wiring, and the battery was so weak that the dome light went out completely while cranking. Put jumper cables on it, and it fired right up, idled fine, ran great. So I am now thinking the FP is good, that it just needs a new battery. Did not have my voltmeter with me, and it was dark, so I couldn't check voltage at pump harness near the tank.
Just wondering if this would make sense to you guys? Most fuel pumps that I have seen die, don't come back to life like that...

Battery going bad can also not give enough power for the electronics causing wildly unpredictable operation. My last one was right at the 5 year mark and it would do all kinds of funny things before totally dieing. See if the battery has a date sticker on it. Usually a round red or yellow with month/year in two digit codes. Over 4 years old is questionable. I usually put it on the charger for 4 hours, then have the battery and alternator tested just to make sure it isn't only one of the items. I've had bad charging systems ruin a new battery when I didn't check both. (You probably know most of this, but threads are read by many that aren't that experienced) I haven't heard of fuel pump problems with this vehicle, one of the things they got right, so I'd check the battery first.