the nitrates are high at the end of a cycle because there has been a lot of ammonia -> nitrite -> nitrate process going on during the cycle. nitrates are removed via water changes, which are not usually done during a fishless cycle, so they build up. that big water change before stocking a recently fishless cycled tank is the key to bring those 'trates down to a better level for the stock you are adding to the tank after the cycle.
what type of filter are you running on the tank? some can be nitrate factories once they gunk up a little. i found i had higher nitrates when using my topfin cartridge style filter, before i modded it with sponges instead of the cartridges.
also, is there a lot of mulm in the substrate? that can lend to high nitrates as well.
as mentioned, test your tap water for nitrates.
yes, you are doing the nitrate test correctly.
i would do water changes daily for a few days, at least 50%, and let us know what the nitrate level is after.