What is happening to Logan?

It's not explained outright in the movie, but the director kind of explains it in interviews prior to it's release (in response to the trailers, mostly.)

It's been known in the comics world for a long time that the adamantium metal in Wolverine's body is poisoning him. The only reason he can survive such a process is because his healing factor is constantly repairing the damage from the metal. In fact, there have been times in the comics when Logan loses his adamantium, and he is notably better off - faster, stronger, heals better, etc. The exact nature of this poisoning is somewhat inconsistent in the comics; in some places, the metal is called out as explicitly poisonous, but in other cases, it's explained as something specific to the bonding process Logan underwent).

In any case, in Logan, the metal is apparently poisonous. Logan says as much to Dr. Rice in the movie, and Dr. Rice does not even try to contradict him:

Logan: He's [Rice's father] the one that put this poison in me.

It's not clear if Stryker and his team knew this when they first applied it, but Zander Rice is at least aware of it's effects by 2029 (though he still chose to apply it to Lara).

Normally, Logan's healing factor keeps this problem in check. But as Logan gets old, his healing factor is starting to weaken. It's still working to keep him safe from the adamantium, but that prevents it from working effectively to heal his other wounds. He still heals, but slowly, and leaves scars. And if he suffers major traumatic injury, he may not be able to heal from it.

This is what the director had hinted at before the release of the movie, though not so specifically:

JAMES MANGOLD: We imagined that it may have when he was younger, but with age, he’s getting older and ailing. Perhaps his healing factor no longer produces baby-soft skin.

Wolverine Adamantium is effecting him AND the corn additives are weakening his healing system. Which has a two fold effect: his body tries to heal the poison from the metal while fighting the effect of the additive. So his system is in constant state of battle. However, once Logan is "dead" and in the ground, his body won't be taking in the additive and his healing factor will repair/kill off additives, resulting in healing factor returning. I feel like he will still be aged because it's not the fountain of youth.

Logan explains it multiple times. Adamantium is the only thing that can kill him. he says this about the bullet, but he also says that "it already is", referring to the adamantium in him and Laura.

Keep in mind the previous Wolverine movie. His healing factor is compromised by a small injected machine. The machine constantly attacks him from the inside, and his healing factor struggles to fix it. It makes Logan weak and prevents other wounds from healing properly, to the point he almost dies. This is exactly what the Adamantium is doing to his body. It weakens his healing factor. It's a poison that's finally overcoming the healing factor.

In the comics, people with Wolverine healing factor, like Deadpool, are basically immortal and can live for thousands of years. But Wolverine cannot due to how much the adamantium affects him. He still lives way past the 50 years of adamantium the movies provide (80s to 2029.)

It might be a combination of two things.

We don’t know why he got this disease. Perhaps the consumption of the said “food additives” may have triggered something or it was just how things turned out for him. Same story for Wolverine. Perhaps the Adamantium alone has been poisoning him or it is the added effect of the food supplements.

It is quite possible that the X-men have been consuming the food additives Dr.Rice has been administering to the world. While the objective is to suppress the X-Gene, mutants may be affected by it in someway or the other. (Like a contraceptive that has side effects for a few)

In case of Wolverine, perhaps the food additives have affected his healing in some way. While his body was able to battle the foreign element adamantium before, with reduced healing, it's not able to battle it as well. This is leading to a steady decline in his healing abilities. Hence he's dying from the "poison" he mentions. X23 only has Adamantium in her hands and feet. She may not suffer the same consequences that wolverine does.

So the combination of food additives and adamantium is causing the decaying.