Logan's healing factor may be getting weaker with age, but it cannot be responsible for his dying because in the Days of Future Past timeline his healing factor was still obviously strong enough to keep him going. What's different? The cure. The US government has been flooding the food supply with GMO corn (among other things) that has a mutant gene cure which doesn't kill mutants, it just edits them out of the gene pool. No more mutant children, no more mutants.
*However* Logan's healing factor may have caused the cure to interact with him differently. Because of his healing factor, and equally important his *using* his healing factor constantly because of fighting and also dealing with the adamantium on his bones, in effect Logan is constantly replacing his body's cells at an accelerated rate. The cure probably weakens mutations over time, but just not enough to really be noticeable in normal mutants: it just does what they said it does: it eliminates mutation from successive generations. Logan's entire body is, in a sense, many generations distant from his previous self. So it is really the cure that is ultimately killing him.
That conjecture is bolstered by the fact that the scientist in charge of the X-23 project freaked out when they had a birthday party for one of the X23 kids. I realized that while it is consistent with them not wanting the staff to socialize with the kids and humanize them, he could also be freaking out about the fact they made a cake for the kids. If the mutant cure is in the GMO corn, it is possible that corn syrup based sweeteners have a chance to contaminate the children and ruin their experiments. Because the cure is probably classified, he couldn't say he was concerned about that fact directly. But it might have explained his anger in that situation.