We just bought a 60g tank a few weeks ago from someone that already had it established and we have had the same problems as well. When you moved the tank and stir up the sand, all the organic matter that has been sitting in the sand gets stirred up and causes a spike in nitrates and possibly anything else that shows more than its suppose to.
Do you have fish or corals in your tank yet? The tank we bought came with a kleins butterfly fish, arrow crab, and a falco hawk fish. The best way to get nitrates down is by doing water changes, but not too big cause you can shock the fish in the tank. Our nitrates are still super high and our fish are still doing great. It was recommended to us that we get some durable corals to help with ammonia levels and other spikes in your tank. We were able to buy a Kenya tree coral, blue mushrooms, green star polyp, and xenias. The carbon media was a good idea. We have bio balls in our over flow as we'll but we can't remove them since they contain good bacteria. Removing them would be like starting with a brand new tank and could stress or kill the fish living inside the tank.
Good luck with your tank!