Explanation
Correct answer: B.
Choice B is the best answer. "Nevertheless" logically signals that the claim in the sentence—that the speaker could visit every state in the US and not find the oldest public library in the Western Hemisphere—is true despite the previous claim about the speaker seeking out old public libraries.
Why the other choices are wrong
Choice A
Choice A is incorrect because "as a result" illogically signals that the claim in the sentence is a consequence of the previous claim about the speaker seeking out old public libraries. Instead, the claim is true despite the previous claim.
Choice C
Choice C is incorrect because "earlier" illogically signals that the claim in the sentence occurs earlier in a chronological sequence of events than the previous claim about the speaker seeking out old public libraries. Instead, the claim is true despite the previous claim.
Choice D
Choice D is incorrect because "in other words" illogically signals that the claim in the sentence is merely a paraphrase or restatement of the previous claim about the speaker seeking out old public libraries. Instead, the claim is true despite the previous claim.