Whenever I turn on the windshield wipers, it immediately blows out the
wiper fuse. If I replace the fuse, it'll blow out again, next time I use the
wipers. How do I go about resolving this? Thanks for your help!

 

At a guess, you got an 19 year old wiper motor that is finished.

This could be simple old age, or HEAPS of use (wet climate), or the PO left the wiper on when the rain stopped (you see that a lot) and that puts a massive drag on the motor and wears its brushes etc real quick.

Something I have done and I DO NOT CONDONE IT, is to put a SLIGHTY heavier fuse in the slot, just to TRY IT. What I mean is that if yours is a 10amp, put a 15amp, JUST TO TRY IT. If that blows, you got motor issues. PLEASE BE CAREFULL WITH THIS FUSE INCREASE, BUT GENERALLY A 5AMP UPPAGE WILL NOT CAUSE ANY DAMAGE.

If this uppage is not something you are comfortable with, find a GOOD auto electrician and let him test the circuits for a dead short.

 

Thanks for the suggestions; Admittedly, I tried putting a
20amp instead of the normal 15amp fuse in, a few months
back when this problem first surfaced -- and it quickly
blew, as well. Right after the fuse blows, when I press the
wiper lever, I can hear a loud repetitive clicking coming
from the wiper module under the passenger side dashboard.

 

OK, sounds like a motor shorting.

Had a thought after a bottle of "nice red wine", haha.

Remove the wiper arm assembly, single nut under the plastic cap. Run the motor and see if it runs longer WITHOUT load then WITH load. If that allows it to run without blowing the fuse, it confirms that the load of the blade on the glass is overloading the motor and blowing the fuse. If it blows "instantly" again, then a short circuit inside the motor is still my best guess.

 

Very helpful comments -- turns out that motor runs
fine without the load of the blade on the glass.