I have a 2003 Ford escape and it kept shutting off when I would leave it running with the Ac for a while. Later on the time my battery light turned on so I figured I needed to charge my battery so I took it over to O'Rileys to get it checked they said "the battery was no good" so I had no choice but to buy a new one. I replaced it and installed my new battery for the same issue to still be there. My battery light is still ON. When I unplugged the positive charge off the battery My truck turns completely off....
I'd definitely suggest your alternator is bad. As was suggested in the comments, you can check the alternator itself at the battery. What you'd be looking for in this case is what the voltage is doing. Put the multimeter leads on the battery while it's running to check the voltage. If the alternator is bad, you'll see the voltage slowly tick down.
If this is a 4-cylinder engine, there is a trick to replacing the alternator without taking the passenger side wheel off and removing a ton of stuff (as suggested in most directions I've read).
The two circled items are captured fasteners which will slide as the bolt is tightened, clamping the alternator in place onto the mounting bracket. Once you get the basic stuff done to pull the alternator, you'll have the two bolts which go through the alternator and attach to these fasteners loose as well. First, don't take these two bolts out. Loosen them so the head is about 1/4" to 1/2" off of the alternator body. Next, use a pry bar to push the head of the bolt back into the body of the alternator (leaving it without a gap). This pushes the captured fastener back and allows the clamping force to be relieved. Next remove the upper bolt. As long as you have everything else disconnected, you should now be able to "clock" the alternator clockwise, which will allow the bottom bolt to pop out of the slot in the bracket and then you can slide the alternator out of its nestled home.
The problem lies with the intake manifold. There is no way to swing the alternator out of the way in order to remove it, you must turn the alternator body so the bottom bolt will be clear the mounting bracket. I don't believe there is enough room between the body and the alternator to pull the bottom bolt out, so this is why the mounting bracket is slotted in the first place. If you don't push the captured fasteners back far enough, you'll get so frustrated with the alternator you'll probably want to shoot the thing.
The V6 version looks much more straight forward to remove, though I've not done it myself. There appears to be three bolts holding it in, which should be fairly easily accessed.