F150 will start but wont run

You can get codes with the key on engine off , things need to be tested if batt is being drained that quick Check all relays fuses

Could be a security issue, some of the older unit would let it run a bit before shutting it down. If the battery is weak then that could cause it. Have the battery load tested.

Have the alternator and battery tested, on these newer 2000 and up vehicles most all problems start with electrical signals to the computer. I had the same problem and replaced the alternator because it was an original part. There were no codes but the customer didn't notice the alternator gauge. New alternator solved all the problems runs like a champ now.

Bring a good battery to auto zoo or the like and they'll tell you it's no good just to sell more batteries , They can not be trusted . I'm not sure who can be trusted

That is why you get an independent shop that you trust too do little things also. If your a good customer sometimes they will just do it cause they know you will be back.