From time to time, I still find it hard to deal with phone calls myself. My advice:
1. You're very very unlikely to talk to this person ever again; you're most likely never going to see this person, either. They are merely an unknown voice on the other side of the phone. In your situation, their job is to help you and to put your needs first.
2. If you slip up in any way - try and make it into a joke (if the other person even heard it, sometimes I talk too softly to warrant saying anything). Lightening the mood instead of increasing your nerves is something that is easier to say / type than do... but once you can, it's life-saving. Try pretending to be your favorite comedian. Mine is Jim Carrey so I'm not afraid to be ridiculous.
However, practice seems to be the best way to become better accustomed to these situations.