OK, let's do some diagnosis:
First, almost all lead acid starting batteries are the same. Yes, there are few different types but within one type I do not care if they are red, or blue, or yellow, or black, or pink. They are all about the same.
Throw a trickle charger (6 amp of so) on the battery overnight (12 to 24 hours) and recharge that battery completely. Then turn on the headlights for 30 seconds to a minute and turn them off again. This gets rid of the surface charge on the cells. Then put a voltmeter across the battery and read the voltage. 12.6 to 12.8 is a good battery. Anything less than 12.6 is a problem battery and 12.6 is marginal.
Next is to figure out if the alternator is charging. Start the engine and read the voltage across the battery again. It should read 13.8 to 14.2 volts. If it isn't then we need to find out why. The alternator could be bad but also the PCM starts and controls the alternator in this truck and it could be failing to start the alternator. Do not ask me why they designed it this way but they did. I actually think I know why but it is speculation on my part.
We could go probe the start line to see if it is active at this point or you could perform a PCM / BCM reset by disconnecting the battery for 1 hour (or more) and go through the above testing again. I would do the reset because it is easier.
While you are waiting for the hour I would suggest disconnection of the ground connections on the left and right inner fenders and cleaning them up until shiny. Then install a tooth washer between the body and the terminals and tighten them up again. These bad grounds cause all sorts of strange problems when they become corroded or rusty.
If after the hour the alternator still does not start then you will need to probe the PCM start line to see if it is being told to start.
Here is the wiring diagram:
I do not know if the wire should go plus or minus. You will need to probe it (the red wire) with the key on and the engine NOT running. Then start the truck to see if it goes to the other state of either plus or minus. The last lead is the field. I 'think' I know what is should read but without investigating it some more, I'll wait to say what it should read.
If all of the above things do not find your problem I would remove the alternator and take it back to AutoZone. They have a machine they can spin it up on and test it out for you. I once went through 5 different alternators from 3 different sources on one Ford Taurus until I got a good one.
One other small problem in this area that is difficult to diagnose is a bad belt tensioner. You can have one of these go bad and slip with no other symptoms other than the battery will not stay charged.
Note: Your battery may now be bad but that may not be the original problem. If a battery is deep discharged a few times it will destroy it or reduce it's capacity severely. If you have a charging problem, get it fixed before replacing the battery.