When I saw the topic of this thread, I immediately thought "Bacterial Bloom" and everything you have said above confirms that a strain of bacteria living in the water column has gone out of control.
I'm sorry. I should apologize ahead of time. My advice will go contrary to what has been said above. I'm not offended if you choose to disagree or offer contrary opinions.
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Water changes typically will not matter because new water has nutrients to feed the blooming bacteria.

(This happened after a water change, right?)
The fast way to correct this is to hook up a UV Sterilizer for just a few days.
The other way to deal with this is described below. Two of the items below are good to do anyway, from time to time, to keep a tank refreshed, vibrant and doing well in the long run.
Don't worry, tanks typically survive these occurrences without any damage.
Moving the coral was okay to do, probably not necessary but okay, especially if the other tank can support them.
A good course of action is to replace the AC with twice as much as usual and then just wait it out. AC can remove nutrients that may have sparked the bloom.
Bacteria blooms will eventually overpopulate, strip the water of the nutrients and die off.
The high Nitrate level is probably due to the large bacterial population in the water. (It takes a lot of bacteria to make the water cloudy.

)
If this were my tank, I would also add a good amount of macroalgae and several scoops of LS from the tank of a friend. The added LS will provide other strains of bacteria to compete with the strain that has bloomed. It may even provide a new organism to eat the blooming bacteria. (Yes, there are organisms which eat bacteria, keeping it in check.

) It's all about balance and equilibrium of the microbial organisms of the biofiltration.
I might also increase the photoperiod, or even turn on the lights 24/7 for a while. The increased illumination will cause all the algae in the tank to compete with the bacteria for nutrients. (I wouldn't scrape the glass because that's good algae too.) This procedure of promoting algae growth is, IMO, better than water changes for dealing with nutrient spikes because it reduces polluting N and P very fast. In my experience large water changes are labor intensive so I avoid them. I'm a lazy reefer.

Large water changes also cause additional stress to tank inhabitants.
Aloha,
Mark
Edited by Mark Peterson - September 20 2013 at 7:02am