Huh? Yes you need to remove the rear axle cover in order to remove the axles. If you do not have limited slip you remove a small bolt, slide out a large pin, which releases the spider gears and then allows the axles to be pushed in so that the "C" clips can be removed with a magnet, then the axles can be pulled out once the disc brake calipers, brackets and rotors are removed.
If you have limited slip (and I would assume you would on a 4x4) the process of removing the axles is more difficult.
IDK if you skills are up to this task. You can find videos on YouTube that should give you a good idea of what's required but if you've never done anything like this before you may quickly be over your head. Plus you may want to consider replacing your limited slip clutches while your in there (as they wear out over time).
I'll look at the link you provided for the F150 and see if the job would be similar/same on your ST. You never answered my question as to whether the '03 ST has a solid rear axle or independent rear suspension.
Edit:
I reviewed your link for rear axle bearing and seal replacement for the F150 and provided your ST has a solid rear axle (like that pictured for the F150) the bearing and seal replacement procedure would be identical.
TIP: If you go ahead with this job, be very careful not to bugger up the head of the small bolt that holds in the large pin for the spider gears and axles in place. If you do bugger/round-off the head of that bolt it will be nearly impossible to remove it. I usually replace that special bolt any time I have to mess with it and use thread locker on it when reinstalling it.