Q: Origin of "dripping with [sarcasm]" 
But the worst part of it was being tried as a . . . criminal, as though his work hadn't had any political consequences. He'd tried hard to set that aside, but the prosecutor hadn't let go, his voice dripping with contempt in his summation —actually worse than that, because he'd been so matter-of-fact in the presentation of his evidence, saving his contempt for later.
Rainbow Six, by Tom Clancy

It is one of my favorite metaphors and it seems to be on the increase in recent years.  When something is soaked to the bone, in both the figurative and literal sense, it is saturated and begins to leak; it drips, hence the phrase in bold suggests the person's contempt was audible enough to actually hear or see it dripping.
It's an effective turn of phrase, but I believe the older expression is dripping with sarcasm, and Google Ngram seem to agree

What really surprised me was how recent its coinage seems to be, the earliest example I found is dated 1921:Arne (in a voice dripping with sarcasm) Well, are you tired of life? Although I did also find a 1913 source that had dripped with sarcasm

Mrs. Pennycook's voice dripped with sarcasm. "Yes, I've been away three years, but I see time ain't softened the tongues nor sharpened the consciences o' some of my old lady friends.

In British English there is a more jocular variant

be dripping with sth
to be wearing a lot of something:
She was absolutely dripping with gold/jewels.

Although, truth be told, dripping with bling appears to be American, and coined in the early 21st century.
Questions

*

*When did "dripping with sth" as a metaphor, first appear in print?

*Is it American or British English in origin?

*Who wrote "dripping with contempt/sarcasm" first?


A: The figurative use of a verbal 'drip' to describe vocalization, that is, a voice, tone or type of speech, appears to have been cliched for nigh on 150 years. I suspect a classical origin, but will not exceed the ELU brief to investigate that.
Also, interestingly, in the earliest uses I have found of 'voice/tone/speech drip/drips/dripped/dripping something', the 'something' that drips is positive (not negative like sarcasm or venom or whatnot). The positive expressions suggest that, rather than a classical origin, the cliche may have simply and naturally transferred from earlier, widespread (and equally cliched) use in descriptions of what are in the main thought to be positive things. 
To avoid belaboring the point with vegetation, nectar and spices ("Dripping with Sabæan spice On thy pillow", Tennyson, 1830; "Piled-up [honey]comb,..which dripped with *amber nectar", Marion Harland, 1857; "Every arch drips with foliations hanging free like lacework", Edward Augustus Freeman, 1849), I will follow the precedent set by your question and only quote "It hangs unclasped and heavy with jewelry, dripping with chain, filigrane, and aiglet" (Harriet Elizabeth Prescott Spofford, 1860; all foregoing quotes from OED).
Equally, the cliche as a reference to vocalization seems to have sprung up in both the US and UK at nearly the same time. For example, 

"It is the true gospel of æstheticism," sighed a "too utterly-utter" disciple of Oscar the First, in a voice dripping with tears of delight.

appears in the 29 Jan 1882 edition of The Courier-Journal of Louisville, Kentucky (paywalled link); and Northampton Mercury of 12 May 1888 (also paywalled) squeezes out

...his speech dripping forth with chilling regularity and precision.

Then, in a story that later reappears in the UK, but first shows up in 1889 in the US, comes 

He at once put his arms close about me, and asked me in a voice dripping with tenderness: "Katie, don't you love me some?" 
The Times (Philadelphia, Pennsylvania), 30 Jun 1889 (paywalled).

The theory that sweet oozing articulation preceded sour, however, may be belied by looser and more creative investigations. For example, this more elaborate metaphor from Liverpool Mail, 6 Feb 1847 (paywalled): 

...until, at length, the impassioned orator, warning Englishmen against being misled by his false insinuations, wound up a tremendous sentence, by calling those insinuations "the dripping venom of a shrivelled adder." An uproarious burst of applause seemed to be involuntarily extorted from all sides of the house.

If such deeper investigations do dash cold water on the theory that the 'drippy' metaphor for types of utterances sprang from earlier, positive descriptions, so then perhaps those investigations may lend more credence to the theory of a classical (Latin or Greek) origin.

A: The term "dripping with sarcasm" appears at least as early as 1892 in a newspaper article about a court case. There is no byline on the piece.



*

*from The Saint Paul Globe February 6, 1892 (paywalled link)


In 1938 the term "dripping icy contempt" appeared in the Chicago Tribune

I can't find any examples of "dripping with hate" or "dripping with irony" as early as the 1892 use of "dripping with sarcasm."
To your second question, My best guess based on newspaper searching that it is American in origin.  As mentioned, the first use I could find was from Saint Paul, Minnesota, and the other early prints I found were in U.S. publications.  Ngram seems to agree that the use popularized later in British English.
Personally, I can't find any uses of "dripping with sth" in the figurative sense prior to the Saint Paul Globe piece.

A: The phrase seems to have originated from "dripping with gore" and "dripping with corruption", the literal uses changing to figurative.  
An example from 1814 is Bonaparte's Last Address to His Friends and Soldiers:

In truth 't was a pantomime— dripping with gore ! 

From Mechanics' Magazine 1843, quoting Thomas Ewbank: 

Narrations of political convulsions, recitals of battles, and of honours conferred on statesmen and heroes, while dripping with human gore, will hereafter be unnoticed, or will be read with horror and disgust, while DISCOVERIES IN SCIENCE, AND DESCRIPTIONS OF USEFUL MACHINES, will be all and all.

Two 09 September 1848 newspaper articles read: 

Never was the truth so dipping with gore; and it is the work of history to wipe out the stain 

A 19 August 1854 article says: 

The book drips with gore 

A 18 March 1872 article says:  

The Drame de Gondo is the title of [?] piece, which drips with gore, and revels in poiniards, pistols and infidelity  

Other than gore, William Kellogg of Illinois speaking 8 February 1861: 

he took a constitution, already dripping with corruption and fraud, covered with slavery of the deepest hue 

Also The Life and Adventures, Songs, Services, and Speeches of Private Miles O'Rielly (1864) has:

There are wires, powerful and numerous, and each dripping with corrupt gold, leading to the site of every contract famous in municipal history.

And in Spiritualism Identical with Ancient Sorcery, New Testament Demonology, and Modern Witchcraft (1866) there is: 

Spiritualism is always the enemy of God...It assaults heaven with the boldest blasphemies, stalks on with the most unblushing arrogance and presumption, and smokes and drips with corruption which would shock the morality of a heathen.

Much earlier and differently there is:

...alludes to Falstaff's entering in a great heat, "his fat dripping with the violence of his motion, as butter does with the heat of the sun." 

quoting from a footnote in The plays and poems of William Shakspeare (1790).  

A: I beg to suggest that JEL's dripping venom of a shrivelled adder is the true starting point. Isn't Dripping with venom a perfect description of an angry rattlesnake or cobra with no literary devices needed?
Expanding Mari-Lou's ngram confirms dripping with venom slightly pre-dates and was sometimes more popular than the other terms. I suggest that was first used literally of snake's fangs, then metaphorically of a person's voice, then expanded sideways to load other voices with contempt, sarcasm or what-have-you.
I also guess the tenderness option follows the same mechanism from a different starting point: waffles or pancakes or some such, dripping with treacle or honey or some such…
