The start of the problem is the NWT's rendition of hemera kriseos in Matthew 10:15; 11:22, 24; 12:36. For some explanation, see here. By "Judgment Day," more properly "day of judgment," Jesus is referring to the coming destruction of the nation at the hand of the Romans. The reference to Sodom & Gomorrah is an allusion to Lamentations 4:6 although the phrasing in Matthew sounds a bit awkward.
By translating the phrase in Matthew as "Judgment Day," and developing a doctrine that differentiates it from "day of judgment" (although they are the exact same phrase in Greek), the Society created this problem of whether the people of Sodom would be resurrected or not. And thus, they waffled back and forth over the years about this, when in fact, it was their manufactured understanding about "Judgment Day" that created the problem, or at the least, kept it going over the years.
Now, as the link above mentions, they never use these verses in Matthew when they talk about "Judgment Day," even though these verses in Matthew are the only ones in the NWT which have the term "Judgment Day."
Concerning Matthew 11:23, Barnes Notes on the NT says, " This has been strictly fulfilled. In the wars between the Jews and the Romans, Chorazin, Bethsaida, Capernaum, etc., were so completely desolated that it is difficult to determine their former situation. " I imagine Josephus has some on this but I don't have time at the moment to look it up.