You have to understand a little bit of chemistry to know if you are using the right product. Sodium thiosulfate neutralizes chlorine & chloramine. When it breaks the ammonia/chlorine bond in chloramine it leaves you with ammonia.
Sodium hydroxymethane sulfonate neutralizes ammonia by changing it to ammonium, a substance non-toxic to fish. Ammonium will read the same as ammonia on most common ammonia tests.
Ethylenediaminetetraacetic acid, or EDTA, is used to bind up heavy metals. This is usually most useful when dealing with well water, but many municipal water supplies will also show traces of metals.
I would bet the product you have does neuteralize chlorine & chloramine, but does nothing for ammonia. This would cause the ammonia spike due to the bacteria which convert ammonia to nitrite being temporarily overwelmed. They are probably converting enough ammonia to nitrite to overwelm the bacteria that convert nitrite to nitrate, giving you a nitrite reading as well as an ammonia reading.