I see some hate mail directed at the AX4N. This is a list of known problems:
Warning: Watch Out for These Ford AX4N Automatic Transmission Problems - ETE Reman - Common Complaints about the AX4N Transmission
One tidbit leapt out at me, that people adding an aftermarket tranny cooler got much better service out of theirs. Maybe most of the problem with the tranny is it just gets too hot? Whatever the case, yours being as old as it is, there's a fair chance that needed upgrades within the transmission have already been done. You may be able to find out by searching its VIN on Carfax.
DO NOT attempt to Engine OFF Coast with an '03 Taurus. You'll cook the tranny. It won't take it.
NOTE: You can go to Motorhome Magazine's annual list of flat-towable dinghy vehicles to determine whether your auto-equipped car can be coasted with the engine off. With 4x4s sometimes the limitation is that the transfer case be in Neutral, but when you're talking about 2wd cars and trucks, it gets more interesting. Sure enough, your Maxx had no mileage limit for flat towing.