Either the test is bad or you didn't do it right. A good liquid nitrate kit will go bad eventually, I've had it happen several times over the years. The strips will never be accurate. So, either way, I'd recommend buying a brand new API liquid nitrate kit from a store with a lot of turnover (so it hasn't been sitting on the shelf for years), because something is wrong.
There's no way your nitrates will go up like that right after a water change, unless your tap water is higher in nitrates than your tank, and since your kit isn't even showing consistent numbers for your tapwater, it's likely a wrong reading.