We need to know the actual test results for the following to before it is possible to make suggestions.
For one, it is not true that you need to do massive wqter changes for any ammonia reading. Whether this is needed or not dependts on knowing three numbers: Total Ammonia ( which is for what most hobby kits test) pH of the tank water amd the temperature of the tank water. A 4th number may be needed if you add salt to a tank and that number is salinity. However, for most FW keepers there is 0 salt involved.
Next. a nitrite reading of 20 ppm will be killing fish pretty fast if remedial actiom is not taking. For one it is also high enough to stall or kill cycle. The normal solution for nitrite in a tank with fish is to add chloride which blocks the nitrite from entering a fish. Salt is Sodium Chloride and will protect the fish until the nitrite is mo longer an issue in the water.
While it is advised ad nauseum that media should never be rinsed under tap water, this danger is highly exaggerated. The bacteria live in a biofilm which offers protection against both chloramine and chlorine. The danger is minimal to the bacteria. And unless the levels of the residual chlorine/chloramine in one's tap is unusually high, it will not be an issue for the bacteria. However, when the discussion is about using water to refill a tank after a water change, it is a different story. You must use dechlor. If you would like a detailed explanation about all of this, I am happy to provide it. It is based on actual science.
The greatest danger from residual chlorine or chloramine is to the fish when there is a lot of water involved as opposed to a small amount that might be left in media rinsed under ones tap and which is quickly diluted when it is returned to the filter/tank and detoxified when it is exposed to dechlor in the tank water.
Despite the above, it is still better not to take any chances unless you are very well grounded in all of this and you are also not going to make any mistakes. Sometimes water companies will push a larger does of disinfectant through the system. This can result in elevated levels for a short time. These levels may not harm humans, but they can harm fish. And then there is the fact that most dechlors also help with certain metals in water and they are a different thing than residual chlorine/chloramine. Unless one knows with certainty what is in their tap water, it is better to be safe than sorry.
I have well water and I never dechlor it. But I do bleach dip things and, when I do that, the next steps are a rinse under my tap and then a few minutes in a bucket with a large dose of dechlor in it.
