Just to be clear, do you mean NitrItes or NitrAtes? If you're talking about healing ammonia burns etc. then I think you meant nitrites. High nitrates would indicate a fully cycled tank.

How high are the nitrites exactly - what's the latest reading?
It sounds like the move to a new tank made the cycle crash for some reason - did you allow the filter media to dry out at any stage? If this is the case, then you need to re-cycle the tank.
20% changes are actually quite small and probably nowhere near enough to deal with soaring nitrites. During cycling, frequent changes of 50% or even more may be necessary to bring the nitrites back down below 1.0. The best way to control your ammonia and nitrites is this:
1. test the water daily for ammonia, nitrites and nitrates
2. if you find ammonia or nitrites have risen above 1.0, change just enough water to bring them back down to between 0.5 and 1.0. This is a safe level for your fish while still leaving enough for the bacteria to 'feed' on and grow.
3. Repeat process daily until no ammonia or nitrite is present, only nitrates. The tank is now cycled and safe for your fish. This will probably take your tank a couple of weeks.
I would remove the Nitra-Zorb from the filter. It is much easier to assess where you are in the cycling process without this present and it will also inhibit formation of the nitrifying bacteria which you need.
I never had much luck with Cycle - won't hurt, but doesn't help a lot either.
The other thing you could do is add a little salt to your tank to help protect the fish against the effects of the nitrites. 1 teaspoon per gallon, dissolved first in a little tank water is ideal. Any salt will do as long as it does nt contain anti-caking agents.
Hope this helps.