The 1889 passages are especially interesting. I'd almost bet that the lady who asked Russell's assistance was L. A. Smith, ca1883.

The reference to Dana makes it pretty clear that he knows those shanties by title only. I'm not surprised that he offers no words.
He obviously doesn't think much of them, and he may not remember them very clearly. He hadn't been at sea for nearly 25 years, and as far as we know he was never a shantyman himself; so the words he'd memorized would have mostly been the choruses. It is also possible that

1. the shantymen he'd known were so vulgar (by Victorian standards) that he'd have felt uncomfortable quoting anything they sang. A serious Victorian frowned on any reference to getting drunk or disorderly, any use of the word "damn" or "bloody," any suggestion of hanky-panky, and perhaps even on a song like "Sally Brown," in praise of a mulatto woman. Minstrel-type lyrics were simply foolishness (Whall pretty much says that's how he felt.) "Uplift" was the watchword.

2. the shantymen he'd known extemporized so freely and unpoetically that the words were just too dull to quote

3. he knew that the inoffensive stanzas floated so freely from song to song that he didn't feel they belonged to any one shanty and so would not be "representative" of any particular song

4. Russell's editor thought that it would be a waste of space to quote doggerel, particularly since the books of L. A. Smith and Davis & Tozier were now available.

Altogether, I don't see anything disingenuous about Russell's failure to quote lyrics, though of course it's disappointing. My guess is that he didn't pay much attention to the shanties in the first place except for the melodies. I'll also guess that the shanties he mentions (except probably those mentioned by Dana) were the most widely sung in his experience.

The "reckless modern" had "hurled" the name not at the sailors or the landsmen but at the songs themselves. "Them" must refer to "these last," which refers to the songs.