NH3 in solution is partially NH4OH, so i guess the tests could overlap. roughly though, to show that there is ammonia in solution, add a few drops of the solution to some copper (II) ions in sol'n, should give a ppt, then add some more of the soln, if there is ammonia, then the ppt will redissolve to give a deep blue solution.
ammonium: heat with NaOH, test gas with damp red litmus.
NO3-: heat with NaOH, test any gases with damp litmus paper, then add devardas alloy, continue heating. if ammonia is released AFTER the devardas alloy is added, then you have nitrate.