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