Deep breath -- it's not as bad as it sounds. In fact, your reef tank will blossom with no fish, as worms and pods and such get a chance to grow their population without predation... and spawn larvae that feed your corals. Also, leaving the tank fallow for 6 weeks will kill off almost any parasites that could harm your fishes, not just ick, in case you have any nasties in there you haven't discovered yet. So you'll start with a clean slate and healthy fishies.
Catching clowns is easy -- just wait until they are sleeping and scoop them up in a jar, specimen container, etc.. If you have other fish, they may be more difficult.
I would do as large a water change as you can manage. Since you are removing the fish, this can wait until you have everyone settled in QT, if that's easier. The high nitrates are a symptom of overall poor water quality, not the ultimate culprit. If you can't handle a big water change, start stepping up your small water changes, 5-10g per day or whatever you can handle. 2 or 3 big changes will be easier and work faster, though.
If the cottony stuff is not hanging off the fish, it is probably a bacterial infection. The infection is probably caused by ick, as an opportunistic infection from the cysts. The heavy breathing is probably due to both the stress of illness and a lot of parasites in their gills.
Here's what I would do:
Prepare QT with all new water, well aged and aerated. Match pH, temp and salinity to the main tank. (You can also use tank water if you want.)
Before catching fish, prepare a freshwater bath. Be sure to match temp and pH to the main tank. You'll want a heater in the bath to keep it ready for you.
After capture, if the fish haven't been too stressed by chasing, give them a short bath in the FW. 5-10 minutes. Stay with the fish with water from their tank handy. If they start to get agitated or collapse on the bottom, take them out right away. (The FW dip will kill off a lot of the parasites and give them some relief.) Clowns will hardly notice a 5-10 minute batch.
Once in QT, give them a day to settle in them start treament with antibiotics. I suggest Furan-2, or, if they are eating well, Jungle Labs Anti-Bacterial Medicated Fish food. The food will be easy and direct, but they have to be eating. Both meds are similar and widely available at Petco, etc.
Daily, siphon the bottom of the tank to remove parasites.
After 5-10 days of the antibiotics, start hypo. (Unless the fish are not better, of course.) Directions here:
http://www.petsforum.com/personal/trevor-jones/hyposalinity.html
After hypo is done, the parasites should also have died out of your main tank and everyone can go back in.
Until you start treating the fish, keep your skimmer running and get lots of O2 in the tank to help them out.