Cycle does not have the right bacteria. Use Tetra Safe Start or Dr. Tim's One and Only.
Does your tap water have ammonia?
If not, then a big water change will reduce the ammonia by as much as the water change. ie: a 50% water change will drop the ammonia by 50%.
What is the pH of the tap and the tank? In water with low pH ammonia is more often in the ammonium form (NH4+) and is less toxic.
The symptoms seem consistent with ammonia poisoning, it is an irritant so the fish will produce more slime coat, and it can irritate eyes, gills and the tender tissue in the fins, leading to respiratory issues and fin rot.
The medications you are adding may very well be killing off whatever ammonia removing bacteria has been getting started.
Nitrifying bacteria does not grow so fast as to cause cloudy water. Heterotrophic bacteria do grow this fast, and there may be other things going on that can cloud the water, for example green water algae would like all the extra ammonia.
Here is how I would handle this:
1) Move any reasonable amount and species of fish to the cycled tank. (Caution: if you are really dealing with disease issues, not just a reaction to the ammonia do not do this)
2) Increase the water changes, as long as the tap water GH, KH and pH is similar to the tank. Keep the filter as clean as possible, and do thorough gravel vacs to reduce the organic load in the tank. If you need to lower the pH add peat moss to the filter, and prepare the water change water with peat moss overnight.
3) Continue using the zeolite. This is starving the bacteria if you use enough to really keep the ammonia down, but I will address this in the next paragraph. Use plenty of it.
4) Keep renewing the activated carbon, to remove whatever medicines you have in the tank. Do not use a blended product. AC cannot be rejuvenated.
5) Use an ammonia locking product, and use the appropriate test kit so that you will be testing FREE ammonia, not the locked up stuff. You will have to look into which test and which ammonia locking products you are using to be sure they are compatible.
6) Add as many live plants as you can, if the light is at least 1 watt per gallon. Anacharis, Hornwort, and several other fast growing plants are good with removing ammonia.
7) Reduce feeding as much as the fish will tolerate without eating each other. Feed as much low protein foods as possible. More vegies, less meat. Read the label on packaged foods for the low protein foods.

Get a product with the proper nitrifying bacteria. You are looking for
Nitrospiros spp. (see the paragraph below the Zeolite paragraph if you cannot find the right product)
9) When you have done enough water changes that the ammonia is significantly lowered use the product. This will mean not doing water changes for a few days to allow the bacteria to settle into the filter media.
How to use Zeolite
1) Make up 3 bags of this material. I use nylon stockings, but media bags are available in stores, too.
2) Day 1: Add 2 bags to the tank. (In the filter, or hanging in the tank near a lot of water movement)
3) Day 2: Swap out one of the bags and put the 3rd bag in. Rejuvenate the one you have removed; read the label or do some research, but I believe this involves soaking overnight in strong salt solution (table salt or other sodium chloride), then rinsing in RO, distilled or clean tap water.
4) Day 3: Put the rejuvenated bag in the tank, remove the bag that has been in there the longest. Rejuvenate this one.
5) Days 4 and beyond: Continue the daily rotation. for a while.
6) When you are ready to add Tetra Safe Start, Dr. Tims' One and Only or other (see below) remove the zeolite. It will only starve the bacteria.
What you are doing: By constantly rotating and rejuvenating the zeolite you are keeping a steady rate of removal going on in the tank. If you need to remove more, add more zeolite to each bag. Gradually the zeolite does get filled with other stuff, and cannot be rejuvenated. This is allowing more ammonia to remain in the tank. This will feed the (hopefully) growing bacteria population.
If you cannot find the proper product in the brick and mortar stores, or on line:
Do the fishless cycle in a bucket.
Set up any sort of circulation through any sort of media. Example: Sponge filter, in tank (well, in bucket) filter, HOB filter such as Aquaclear (Square sided container such as a storage box is great for an Aquaclear)
Add a little media from the cycled tank. I would want to add as much as you can, but do not take so much from the cycled tank that it starts to have problems, especially if you have moved a few fish over to it. A handful or two of gravel would be fine.
Add ammonia (clear, non sudsing, no perfume) so the test reads 5 ppm.
Keep testing and add more ammonia as needed to maintain 5 ppm. After a few days test for nitrite. When nitrite shows up allow the ammonia to drop to 3 ppm, then maintain 3 ppm. Monitor the levels of ammonia and nitrite. When the bacteria have grown a big enough population that they can reduce the ammonia and nitrite to 0 ppm overnight add the filter to the troubled tank.
38 tanks, 2 ponds over 4000 liters of water to keep clean and fresh.
Happy fish keeping!