Yea but a 13 f250 can have 20 or more computers, and most of them can power up even with the key off for different reasons. My 16 6.7 250 has the ecm, tcm, transfer case cm, air bag, abs, instrument cluster, steering column control module, drivers door module, 2 nox sensor modules, passenger door module, 2 modules for radio and I don't have the fancy Sony one, body control module, those are just the ones I can think of off the top of my head.
Do what
@a_kelley said above. If the draw goes away it's the alternator, if it doesn't you're going to have to start pulling fuses til it does basically. That will narrow down which circuit it's on.