Back-dating "drunk driver"

Solution 1:

I have this dated 1787, culled from the online Felix Farley's Bristol Journal (Bristol, England), Saturday, November 3, 1787; Issue 2036. 17th-18th Century Burney Collection Newspapers [accessed 2018-12-11].

Solution 2:

The crime (or misdemeanor) of drunk driving—and use of the term drunk driver to characterize an offender guilty of it—evidently goes back in some municipalities to the days of the horsed carriage. From "Police Courts: Friday, July 22: Perth," in the [Perth, Western Australia] Inquirer and Commercial News (July 29, 1898):

A Drunk Driver.—Charles Hillman was charged with having been drunk while in charge of a horse and cart in Hay-street. The accused, it was stated, was sent out in the morning by his employer, Mr, Gullen, to deliver a round with a baker's cart. He, however, got hopelessly intoxicated, and neglected his master's property and business. The accused was sentenced to 21 days' hard labor.

Application of the term drunk driver to the impaired driver of a horseless carriage appears at least as early as 1921. From "Notes and Comments," in the [Barcaldine, Queensland] Western Champion (January 1, 1921):

There is one magistrate in England at least who thinks some of the laws require revising and bringing up to date. For instance he considered he should have greater power for dealing with drivers of motor cars who have indulged too much in the cup that cheers, but also inebriates. While dealing with a case the other day he complained that "It is ridiculous that I can only fine the drunk driver of a deadly thing like a motor car the same as a drunk driver of a donkey cart."

To see the original articles from which these excerpts came, simply click the linked article names.

As Hot Licks notes in a separate answer, the phrase drunk[en] driver goes back considerably farther than the late 1890s. The earliest instance I found in a Google Books search is from a letter to the editor written by G. Cumberland of Bristol, England, dated October 18, 1814 touching on the regulation of stage-coaches, printed in The Monthly Magazine (April 1, 1815):

As to the expenditure of constitution, that or the risk of a broken neck is not to be expected to be thought of, by a being, who imagines he exalts himself in the world's opinion, by suffering a drunken driver to extort money from him at every stage, under penalty of being insulted with foul language, (yet knows the fellow is amply remunerated for his services by the coach-owner;) and as often repels the travelling vagrant on foot with that harshness and unfeeling pride, that brings the blush of indignation from the wounded heart.

This is the same instance that Hot Licks identifies as "Drunken driver from 1815."

The instance of "Drunk driver from 1848" in Hot Licks's answer is actually from an entry in Walter Scott's diary, dated May 28, 1828, reprinted in J.G. Lockhart, Memoirs of the Life of Sir Walter Scott, Bart., volume 9 (1848):

Our elegant researches carried us out of the highroad and through a labyrinth of intricate lanes, which seem made on purpose to afford strangers the full benefit of a dark night and a drunk driver, in order to visit Gill's Hill, in Hertfordshire, famous for the murder of Mr. Weare.

The earliest instance that a search of the British Newspaper Archive turns up is this one, from "The Fleets," in the Hull [Yorkshire] Packet (May 21, 1805) [combined snippets]:

On Saturday night, the Leeds True Briton coach, was overturned coming from Leicester, through the carelessness of a drunken driver, who turned it out of the road where no impediment existed to interrupt his progress. There were five inside passengers and several outside, all [?] of whom were much hurt.

These three instances, of course, involve drivers of horse-drawn conveyances.