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In the world's best text editor, the traditional way (as far as I'm concerned, at least) of getting paragraphs of plain text to wrap is to use autofill-mode. That's what I'd been doing.
This mode has a problem in that it inserts a hard break at the end of each line. This wouldn't be a problem except when you are interacting with the non-Emacs world and with such productivity applications as web browsers, email programmes, word processors and so on.
To fix this, the solution is to use longlines-mode. This mode makes it look as if your text has a linefeed inserted at the end of the line, but in fact, when you save the file or copy it anywhere, that linefeed disappears.
To do this, I commented out the line that turned on auto-fill:
(add-hook 'text-mode-hook 'turn-on-auto-fill)
and turn on longlines mode:
(add-hook 'text-mode-hook 'longlines-mode)
Hopefully, that should do the trick. More info.