The narrative pretense of continuity is about the broad strokes, not the exact details.
That's sort of the way I treat the overall Star Trek continuity. As a generalized story. If you sweat too many of the details you'll drive yourself crazy. And some things you can sort of handwave away with some minor explanations. Khan knowing Chekov from TWOK even though Chekov wasn't on the show yet for "Space Seed" for instance---it's easy to envision Chekov being assigned to some other area of the ship and encountering Khan. As much as I harp about continuity sometimes--and set design---I don't sweat all the details.
First the Eugenics Wars which was implied by TOS to be WW3 in the 90's. Killed tens of millions. They ignored it in a couple of 90's time travel stories in DS9 and Voyager.
Then it was it replaced with WW3 in the 21st century that killed 600 million
Guys, Greg Cox already outlined the history of the Eugenics Wars....duh

In all seriousness, yes, "Space Seed" did combine the Eugenics Wars and WWIII. The first mention of WWIII as a separate war was probably "Bread and Circuses" (though it could still be interprester as the Eugenics Wars I inferred it to be a separate conflict). And I'm trying to recall, was WWIII mentioned in "The Savage Curtain"?
First Contact suggested humans started eradicating poverty and social problems after the Vulcans came, but years earlier in TNG, it said humans were still desperate and struggling almost 100 years later.
When was that said? If you're thinking of "Up the Long Ladder," the line there was that Earth was recovering from the war in the early 22nd century, not that anyone was desperate and struggling. A recovery process from such a large global conflict could easily take 6-8 decades to be fully complete.
And the line in "First Contact" about eradicating all the Earth's ills I don't think happened overnight. First Contact was probably just a first step of many. Even on Enterprise there are times when you get the sense that Earth only turned a corner in the previous 40 to 50 years. It's easy to imagine recovery taking to the early 22nd century.
Part of the charm of the tie-ins in the first few decades was how different they were from each other, how many intriguing alternative perspectives they offered.
Yeah, I have to admit it's interesting when reading an older novel. In some ways those early novel writers probably had some more freedom because there wasn't as much canon. Writers today have a lot more to work off on one hand, and a lot more resources to turn to, but at the same time I imagine it sometimes ties their hands on certain things.