Last time I checked, sears hardware store (and likely many other places) have a free, while you wait, water test. Go into the store and ask, they will give you a small jar and instructions on how to fill it up (let water run for X number of minutes) take back the jar and they will do a basic test and then recommend a water softener. You could also call the culligan man or some such water treatment service. My guess is they will offer a free test and a high pressure sales along with it.
also, asking your neighbors is a good idea. if no one else has an issue then something else is going on. Could something be lodged in the heating element that is melting and burning when the DW is in use?
let us know how you make out.