I'd say it's never too early to book for Thanksgiving. It's the busiest travel weekend of the year, so I suggest you book soon to avoid getting shut out.
I've had good luck with Orbitz and SideStep, so check those to see what's out there for the dates you want to travel. Or ask your travel agent to watch fares for you. It's unlikely that there will be a huge fare sale for that particular week, so if a fare looks good now, book it. I'm sure you can beat NCL's $500 fare.
Also, check alternate airports. You may prefer LAX, but you could also go out of Orange County, Burbank, Long Beach, etc. And you can also check Fort Lauderdale as well as Miami for your destination.