Most has already been suggested, but these are also my suggestions:
I would use a water conditioner if you aren't exactly sure what your filter removes/changes. I personally use AquaClear and/or AmQuel +.
Stay away from the AmmoLock type chemicals if at all possible as some of them can really disrupt the cycle by stalling or killing bacteria. AmmoLock works by binding the Ammonia to a less harmful state. I believe AmQuel + and other products work the same way. It makes it less harmful to the fish, but keeps it in the water. It will still read (possibly incorrectly) on test supplies. It may also stall the cycle, or at least slow it down, though most products mention that it will not interfere at all.
The BioSpira should be ok for fish. It's simply cultured bacteria (and maybe something else).
Do frequent
PWC, but not more than 50%. Perhaps 25% to 30% is best. Any more may stress your fish. HOWEVER, if you are left with a decision of having to do a massive water to drop the levels while stressing your fish, or the possibility of your fish not making it, then do the massive change (duh! lol).
It may be too late, but try to pull over any of the old substrate/deco's in the other tank. If you're lucky, you may have some bacteria still alive (though I doubt it). It wouldn't really hurt to try.
Water doesn't cycle.... not in the nitrogen cycle anyway. Water cycles in it's own cycle and doesn't directly help your Ammonia, Nitrite, Nitrate, etc, By changing the water in a
PWC, you remove parts of the harmful substance, but not all of it. In some cases, a
PWC may remove more of less than expected. Since the substance is diluted in the water, it should be an even amount no matter the change, but there is always a possibility. The only way to rid the water of all harmful levels is to change all of the water, but for obvious reasons, we can't do this.
With my experience, Ammo chips only work for a couple of days or so. I had an experience (controlled, to see what would happen) where I got a spike of 5
ppm. Adding Ammo Chips dropped it down to 1
ppm over the course of 3 days. On the fourth day ammonia was back up to 4
ppm*
*Experiment contained no fish, rather a controlled addition of 5 drops of pure ammonia per day in 30 gallons.
With my experience, carbon (
AC) didn't do anything. Using the same setup experiment above, ammonia stayed at 5
ppm with a large dose of
AC added. Is seems that
AC just removes other substances, but not anything from the nitrogen cycle. Perhaps it was just my experiment that failed though.
Your fish may have died just from the shock of the move or from other semi-natural causes. With 2 deaths of the same type, it would appear to be a problem somewhere though. Take a look at the ideal conditions of all your fish and find out if there are any major differences (like pH) and test for that. It may be that, (using pH as the example) your pH changed drastically and your sharks couldn't handle it, but it still falls in the range of your other fish.
These are my thoughts. Most have been stated. Your nitrogen cycle levels seem low enough to to kill, but they still appear out of whack and should be handled as quickly as possible.