You are good some good advice here, which doesn't surprise me.

But you may still want to post the same question over at
rec.autos.makers.chrysler
rec.autos.tech

The folks over there may offer more experience with this particular
problem. You may want to cross-post to those two groups. You
may also cross-post to alt.home.repair & sci-electronics.repair
since some of us are interested in the opinions on this topic.

I've seen discussions before on your problem, which appears to be
well-known and common. You'll want to check the resister block
PLUS all electrical connections between the fan, resister block and
the blower motor. Sorry, I don't remember more on this issue since
I don't follow the auto groups on a regular basis. I do remember
that the "high" fan setting is usually the last to fail since it is the
setting which doesn't depend upon any resistors to limit current.

You could also try a "Google Groups" search to check on past
newsgroup discussions of this problem.

In your situation, I would:
1) Research via newsgroup questions and Google searches.
2) Clean switch contacts and reapply grease. I'd use a bit of PERC
on a Q-tip to remove grease from contacts. PERC is available
in some versions of electronics cleaners. It is also available
in the "chlorinated" versions of brake parts cleaners (on sale
currently at Car Quest for $1.59 per can). I'd clean contacts
with a bit of Lime-Away on a Q-Tip, followed by a final cleaning
with PERC on a Q-Tip. (I buy cans of "chlorinated" brake parts
cleaner by the dozen when it goes on sale).
3) If possible, check ohms on every resistor in the resistor bank.
Verify these readings against a published reference. Or go
to the dealer's parts department with your ohm meter to get
some reference readings.
4) Clean all electrical contacts in the switch-resistors-fan circuit.
Apply good grease to all contacts before reassembly.
5) If this doesn't help, I would also attempt to "jumper" across
switch position #2 to see if that runs the fan. If so, then it
would identify that the problem is in the switch. If not, then
there is a problem elsewhere - a bad contact or a bad resistor.
Always attempt to eliminate variables in this manner.

Good luck,
Gideon