John, I've received an image of "The Rover," and indeed the printer used the "long s" as appropriate throughout.

But according to Wikipedia the Encyclopedia Britannica was using the "long s" as late as 1817.

"Gallows" in "gallows old whore" appears to match Oxford's sense 1, Fit to be hanged; villainous, wicked," which goes back to the fifteenth century.

But in "gallows old shoulder" it's more like sense 2, a mere intensive, which Oxford dates only from 1789. (Hence Legman's date.)

So the actual date is uncertain. My own guess would be somewhere between 1785 and 1820, with "ca1800" being a good compromise.