No, he progressively loses his sanity
Although Arthur does kill increasingly more people, his mental condition seems to improve as time goes on and his uncontrollable laughter even stops.
Call me crazy (ha ha), but I would qualify a mass murderer as more insane than someone with emotional incontinence.
As you mentioned, Arthur, at the beginning of the movie, has neurological disorders (uncontrollable laughter) and is socially awkward. That's it. Otherwise he is pretty much an average guy. This is intended so that the audience can relate to the character, and feel pity for him because of how unfair life and other people are being to him.
Arthur becomes clearly more competent socially, being able to lie to the police officers
Arthur has lied before, in fact in the movie he lies to his boss right before the train scene:
Boss: Arthur, I need to know why you
brought a gun into a kids hospital.
Arthur: It's a part. It's part of my act now.
If you start from the definition of sanity as behaving abnormally, then Arthur, on his road to become the Joker, clearly goes more and more insane. Not only does he kill people, his murders go from self-defense (kinda), to cold-blooded execution on public TV.
This is him talking to his mother for the last time:
Arthur: You know what's funny?
You know what really
makes me laugh?
I used to think that
my life was a tragedy,
but now I realize
that's a fucking comedy.
He doesn't stutter, or laugh uncontrollably, that doesn't make what he says less crazy. Here is him talking to Murray after confessing his murders:
Arthur: I've got nothing left to lose.
Nothing can hurt me anymore.
My life is nothing but a comedy.
Murray: Let me get this straight, you think
that killing those guys is funny?
Arthur: I do.
Arthur does manage to speak much more fluently on Murray's show than when he did his one man show, which was a similar exercise of being in the spotlight and talking to a crowd, but his newly found confidence is a consequence of his growing insanity: he doesn't feel stress and anxiety as he used to because he doesn't care about the consequences of his actions anymore. What you are referring to as him becoming more sane is actually his growing confidence.
Finally, in the news at the end:
The police led the suspect handcuffed out of the studio
...when asking him what he did...
...merely just a punchline to a joke...