You're right to wonder what they're talking about.
Here's what they're trying to point out. In this sentence:
Now that I'm married, I don't go out in the evenings so much.
the word now introduces a subordinate clause: "I'm married". The phrase now that functions like "because" (also a conjunction). In British English, people sometimes drop the that, but I (native AmE) find that bizarre and hard to follow.
In this sentence:
Now I need to pull my pie out of the oven.
the word now only indicates the time of the main verb, "need". It doesn't introduce a subordinate clause like "because", "since", etc., nor connect coordinate elements like "and", "but", etc.
Now, about those other two sentences. There is a way of using "Now" at the start of a sentence to indicate a transition to a new topic. I used it at the start of this paragraph. Should we understand that kind of "now" as an adverb, by analogy with "therefore", or as a conjunction that introduces a whole sentence, like "So, …" or even "But…"? There are good reasons for both.
And that's my answer: there are good reasons for both. That kind of "now" functions like an adverb and also like a conjunction—at the same time. Sometimes one role is more prominent than the other. That's it. That is the whole truth. There is no need to categorize that kind of "now" as just one part of speech or just the other—unless you're writing a dictionary, grading exams, or just obsessed with categorizing everything into exactly one of some fixed set of categories. The notion of time is present in "Now that I'm married…," so you could say it's an adverb as well as a conjunction. Even "therefore" is reasonably considered a conjunction as well as an adverb. Forcing overlapping cases like these into one category or another just leads to semantic arguments.
Some authorities call this kind of word a "conjunction or adverb"; others call it a "conjunctive adverb". Those sound to me like reasonable ways out of the forced choice. If you're studying for an exam, though, you'd be wise to find out which arbitrary categorization the grader adheres to.