I just called Dell back, and although I had to explain the situation five times, the guy I talked to did what he was supposed to do, and it only took 10 minutes.  There was no stupidity on his part, and he told me specifically that they'd be sending the parts to the tech guys, not to me.  I now have a new dispatch number, and I'm just waiting on the contractor to call me to schedule a time to do the repairs.

Whatever the case, it was still a clusterfuck.