You'd be surprised at how much power it takes to turn an alternator, especially under full load.
Not really-
As I said, a 135A alternator(which at the higher end of the middle range of what I could find for F150 alternators-anything from 95A to 150A) can output 2000W, although that's peak output and most alternators need to be spinning ~4K+ to hit peak(and when they get there will depend on the relative diameters of the crank pulley and alternator pulley).
Let's assume, though, for the sake of discussion, that it's actually at peak output because of the battery. Conveniently enough, we can convert watts directly to horsepower, and 2000W is ~2.6HP. Of course there's some loss from the bearings and such, and I'd guess it's at least a little bit load dependent, but an unloaded alternator isn't exactly difficult to spin by hand. A dirty vehicle may well add enough air resistance to rob that kind of power(if not more) at highway speeds.
Just spitballing, but to bring numbers back to reality I'd guess a less than great battery(not going to call it "bad" because the OP just called it old, and didn't indicate that it was having issues consistently starting the truck or anything like that) might at most draw 10A more off the alternator than a "good" one. An alternator putting out an extra 10A at 14.4V will be using 144W, or about .2HP. Driving in the rain at night with your lights and wipers on likely pulls as much if not more power.
There are things that make real, noticeable differences in gas mileage. Underinflated tires are a big one, and experience across a few different cars has told me that I can easily lose 1-2mpg when they are low. Aside from the safety fall out from the Explorer/Firestone fiasco, one of the other stated benefits of requiring TPMS in 2006(or was it everything made after 2006? Whichever it was) was that "most" people didn't know when their tires were low and it could end up having a big environmental impact.
For the OP, if they really wanted to save gas, going back to standard size wheels and street tires and saving the off road tires for actual off road use would probably do a whole lot more for their gas consumption than changing batteries.
