Carolyn,
We flew AP NAN-LAX a few years ago, not great. Excuse me for butting in but you might want to look at breaking your trip at Nadi and spend a couple of days in Fiji. Below is a Q&D I pulled off of Travelocity. I don't know how you feel about spending a couple of days in Fiji but I've done lots worse things in my life. Again please excuse my presumptiveness.

Greg


Fri, Dec 3, 2010Auckland International Airport (AKL) to Nadi (NAN) Depart: 01:00pm

Arrive: 04:00pm
Auckland, New Zealand (AKL) to
Nadi, Fiji (NAN)
Qantas Airways
Flight 3843 operated by Air Pacific (on Boeing 767-300)
Adult fare rules

Total Travel Time: 3 hrs

--------------------------------------------------------------------------------

Thu, Dec 9, 2010Nadi (NAN) to Austin Bergstrom International Airport (AUS) Depart: 10:50pm

Arrive: 12:05pm
Nadi, Fiji (NAN) to
Los Angeles, CA (LAX)
Qantas Airways
Flight 3839 operated by Air Pacific (on Boeing 747-400)
Adult fare rules

1 Stop – change planes in Los Angeles, CA (LAX)
Connection Time: 6 hrs 55 mins
Depart: 07:00pm

Arrive: 11:50pm
Los Angeles, CA (LAX) to
Austin, TX (AUS)
American Airlines
Flight 2246 (on Boeing 737-800)
Adult fare rules

Total Travel Time: 20 hrs