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If nitrate is raising slowly, it means both bacteria are working as long as other 2 stay at 0. Maybe something is eating up the nitrate, but it's ok if they stay 0 over a week.
How are you introducing the new fish to the tank? Drip acclimatising them is the best way to do it. Floating in a bag and adding tank water periodically is okay. It would explain it if you've just been adding them straight to the tank as is.
What temperature is your water at? What sort of temperature were they in in the shop?
Since you've got 0 ammonia and 0 nitrite (and assuming these readings are correct), it seems you've got a cycle going, so that wouldn't appear to be the problem here. I'm just shooting in the dark to try and figure out what the problem is.
If there are no fish at all in the tank, the ich will be gone in less than a week. The ich parasite only lives for a short time in the substrate then it must find a host to continue its life cycle.
Since this thread is over 4 days old, since you lost our last fish, you could theoretically restock right now with no problems. I think I would approach things more judiciously. You still do not have the "smoking gun" that killed off your tank so try instead doing a huge, over 90%, water change. At that point you should have no remaining contaminants in any quantity in your water. We only measure nitrogen compounds but they are far from the only thing in our tank water. We simply do not measure the rest. The 90% water change is intended to be a reset for what you cannot measure. After that, try out a small group of zebras in your tank. The temperature should be held low, maybe 23 to 24C, since zebras do not do well at high temperatures and see how things go.