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I'm using the "Read Aloud: a Text to Speech Voice Reader" for Firefox. When I get hit with a paywall, I refresh, and before the page fully loads I click the reader.
It grabs the whole article. You can let the reader read it, or just scroll through and read it manually-- all of the text is captured.
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Update: the NYT must have figured out that text-to-speech can bypass the paywall, because now their articles offer "Listen to this article" as a native option, and it does in fact also bypass the paywall. Apparently it is difficult to get around this, so they decided "if you can't beat em, join em"