you're aspose to get drowzee and he will put you in the nightmare. you will find Drowzee at Mt. Something (not the actual mountain name. I forgot it!!)^_^
I haven't seen azurill in Pokemon Mystery dungeon explorers of time a lot.
I gues you could find azurill in one of the water related dungeons. (Example: Waterfall Cave, Brine Cave, etc.)
You just find it in the left board, saying something like "Help me find the Mystery Part!" (Easiest way to make mission appear is with wonder mail. Just google "pokemon mystery dungeon explorers of...
The mystery part is an item that lets you see legendaries wild in dungeons, but they can only be found on specific floors of certain dungeons....for example: Celebi can only be found on floor 10 of...