A major hint that Azazel survives in the end is the movie itself. Azazel has the ability to recall any memories given to a person he hosts. So, that would explain the moments we see Azazel try to take over Gretta's body. That couldn't possibly be a part of Hobbes's memory because he wasn't there at the time. In other words, the whole movie was Azazel recovering memories of Hobbes and some of his own.
The movie demonstrates that people Azazel possesses don't recall their actions while under control. Think back to Reese at the beginning, trying to possess Hobbes. Had he succeeded, the real Reese would have returned to his own consciousness minutes before his own execution for crimes he has no memory of committing.
This happens anyway, since he possesses a nearby guard instead.
No, he leaves the body after Reese's death. Anything shown from Azazel's point of view is done with the grainy yellow filter and the odd background noise. He floats out of the body once he is dead, and THEN enters the guard.
Don't even think about all the mass murderers and serial killers that were proabably demonically possessed in the film's universe.