No, his healing factor is not working as before
As explained in the official screenplay:
Now might be a good time to talk about “fights” described in the next 100 or so pages. Basically, if you’re on the make for a hyper choreographed, gravity defying, city-block destroying, CG fuckathon, this ain’t your movie.
In this flick, people will get hurt or killed when shit falls on them. They will get just as hurt or just as killed if they get hit with something big and heavy like, say, a car. Should anyone in our story have the misfortune to fall off a roof or out a window, they won’t bounce. They will die.
As for our hero with his so-called eternal life and healing. Well, he’s older now. If you keep reading, you’ll discover Logan’s about to get his ass kicked. But before we get to that, we should make it clear his abilities ain’t what they were. Yes, he’s a drunk, but he’s also fading on the inside. Adamantium implants leeching into his system, causing chronic pain and diminished healing, hence booze as painkiller.
So by all means, go ahead and worry about him. Now where were we? Oh, yeah--
- Mangold et al. (2016), "Logan", Final Shooting Draft Screenplay
He's older, and the Adamantium plating of his is slowly poisoning him.