Your guess seems correct
In addition to the information you posted, we have this little piece to work with...
Personality. Azuredge is sworn to protect Waterdeep, and it desires to be wielded by a law-abiding person willing to dedicate everything to the city’s defense. The weapon is patient and takes its time finding its ideal wielder.
We also know that
Azuredge is actively rejecting Meloon (or, more accurately, the intellect devourer in Meloon's head).
Each day at dawn, Meloon engages in a telepathic contest of wills with his magic axe, Azuredge, before leaving his room at the Yawning Portal. The axe wants a new wielder, but Meloon refuses to part with it.
This is a dead-ringer for someone having a Conflict with a sentient magic item. I'm not going to copy the entire chunk in here, but here's the important bit summarized.
If you lose your contest of wills with a magic item it can make demands of you, such as demanding you never part with it, pursue its goals, part with some item it hates, or--importantly--give the item to someone else. If you refuse, it can:
- Make it impossible for its wielder to attune to it.
- Suppress one or more of its activated properties.
- Attempt to take control of its wielder
We know that
Meloon has lost at least one contest with the weapon, because his stat block says "Meloon wields Azuredge (see appendix A) but can’t attune to it, and thus gains none of its benefits."
As for the why of things, the book does not address this directly but it is likely a safe assumption that it is looking for a new owner because
it's old owner is effectively dead, and his body is being driven around by the minion of a horrible aberration who works with a crime lord. And said minion is trying to use Azuredge in service of said crime lord. See, Intellect devourers don't 'control' you. They eat your brain and replace it with themselves--"you" are no longer in there except on a very, very thin technicality (punt out the devourer then immediately hit the body with Regenerate can restore you without resurrection magic).
Suffice to say...this is pretty antithetical to the weapon's personality, and explains what is going on.
However, if you have a look below Meloon's statblock, you see this little bit of info (again, condensed because spoiler tags protest to being multiline)
If Meloon is killed and raised from the dead, his true self is restored and his statistics change as follows: [...] He can attune to Azuredge
Thus, it is reasonably clear that if the current state of affairs is resolved, the conflict is resolved as well, and Meloon can carry on wielding his axe. After all, if he can re-attune to the axe, it has stopped rejecting him. Naturally, as DM, you may opt to ignore this if you want someone else to end up with the axe. (Note: As a DM, I advise caution at handing Azuredge to a player long-term, especially at low level...that axe is a friggin monster).