There are numerous threads on FT and the AA subforum on (saver) awards USA to Australia. Premium seats are *** harder to get and at a peak time is **** times harder to get. On your dates the airlines will be expecting to sell every seat for top dollar, especially 4th Jan. You will likley need so split your party and be prepared to travel on different days/airports/routes/airlines. Do not fixate on LAX & SYD as thousands of others do (Look at SFO DFW / BNE MEL and more).
The award booking link is practical advice.

AA subforum links (this thread is likley to get merged into one of them)
http://www.flyertalk.com/forum/ameri...ssistance.html
Oneworld and other airline Awards Rules, Information 2015 on
http://www.flyertalk.com/forum/ameri...er-thread.html
http://www.flyertalk.com/forum/ameri...solidated.html
http://www.flyertalk.com/forum/ameri...solidated.html
http://www.flyertalk.com/forum/ameri...demptions.html

Airlines that fly USA/Canada to/from Australia (not from all airports to all airports)
- American Airlines AA (LAX - SYD) [OneWorld Alliance] (AS partner)
- Air Canada AC (YYR – SYD, BNE from Jun 2016) [Star Alliance]
- Air New Zealand NZ (LAX SFO IAH YVR via AKL) [Star Alliance]
- Air Tahiti Nui TN (LAX via Tahiti. Last segments on codeshares) (AA & DL partner)
- Delta DL (LAX – SYD) (VA & AS partner) [Skyteam Alliance]
- Fiji Airways FJ (LAX HNL – BNE SYD BNE via Fiji) (QF, AA & AS partner)
- Hawaiian HA (HNL – SYD BNE) (AA & VA partner)
- Jetstar JQ JQ (HNL – SYD BNE)
- Qantas QF (LAX SFO JFK DFW HNL YVR*– SYD BNE MEL) [OneWorld Alliance] (AS partner) {* seasonal}
- United UA (LAX SFO – SYD MEL) [Star Alliance]
- Virgin Australia VA (LAX – SYD BNE) (DL partner) MEL LAX restarting 4 April 2017
And others via Asia & Middle East