Gee whiz.  I've not read any posts that I can remember about anyone having
any serious rear-end problems on any Explorers.  You might want to do a
Google search and see if anything turns up.  I can't seem to remember
exactly what the sympoms are of not adding the limited slip additive to the
rear differential but what you described might fit.

My experiences with my local Ford dealer have not been real great.  I took
my '91 in for a brake booster recall and asked them to check the front
locking hubs.  They determined that the hubs were bad and it would cost $950
to replace them.  They also found about $3000 worth of other stuff to fix.
I might have believed them except they also said I needed new radiator
hoses.  The hoses on there were only about 2 weeks old.  This discredited
anything they told me from then on.  I replaced my auto hubs with manuals
for about $165 or so.

The bottom line for me is that the Ford Service Department seems to be in
cahoots with the Sales Department.  If you have a car worth, say, $5000 they
will find $7000 worth of repairs it needs so you'll just go buy a new one
instead.

If your tires are worn you need new tires anyway (probably).  I just find it
hard to believe your rear end is already worn out.  Personally I would
change the oil and add the additive and see how that goes.  Did you check
the oil level?  You just need (probably) a 3/8" drive rachet wrench and a
finger.  You remove the plug, stick your finger in there and see if it's up
to the plug.  Maybe it's just low.  To change the oil you normally remove
the cover and use black high-temp silicone as a new gasket.  This can be
difficult sometimes.  The other way is to get a gear oil pump and pump it
out but you won't get all the oil out and you can't take a look inside and
inspect for wear etc.

I you get a Hayne's or Chilton's manual it *might* tell you how to remove
the transfer case sensor.  They seem to be overcautious when it comes to
transmissions and related stuff and figure that no information is better
than a little.  You can probably find a Ford Service Manual on CD on eBay
for around $10.  That will tell you how to remove the sensor but it takes a
while to get used to finding anything on those CD manuals.

Personally I'd get a couple more opinions before having the differential
work done.

Definitely get some other opinions and estimates. I needed the
differential rebuilt on my '94 when it had only 50,000 miles on it.
Possibly ruined by backing too far into the lake while launching my
boat. The dealer wanted some similar price to what you were quoted, I
had it done at my local mom and pop shop for $600 for a complete rebuild.