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The colloquial phrase to spit feathers has two distinct meanings:
– the earlier meaning is: to have a parched mouth, to be extremely thirsty;
– the later meaning is: to be or become extremely angry or vexed.

However, those are perhaps two distinct phrases, of different origins, which happen to be identical in form.

The following is about the phrase to spit feathers used in reference to being parched:

1-: Origin:

Of American-English origin, the phrase to spit feathers is probably based on the image of a thirsty person whose mouth is so parched that it feels as if it were filled with feathers and that the person thus feels choked.

In fact, in variants of this phrase, the verb spit is followed by nouns designating other dry choking substances (especially cotton) instead of, or in addition to, the noun feathers.

These are, for example, three of those variants, in chronological order:

1.1-: From Letter from an agent of a mercantile house in Liverpool, in reply to his employers, by ‘Roger Botherum’, published in the Louisiana Gazette (New Orleans, Louisiana, USA) of Thursday 11^th October 1821 [page 2, column 3]:

I felt the loss of my pistareen, (which happened to be a true Spanish one and of course in demand by the good brokers of Boston, the Judea of New England) inasmuch as it would have procured me a dozen of noddies’ eggs, or a few lemons to make a gargle for my husky throat, as I am wont, in this sultry season to spit cotton wool by the bale.

1.2-: From The Well Conducted Farm, about a farmer in Massachusetts, USA, who was keeping his employees from drinking alcoholic spirits, published in The Temperance Advocate, and Australasian Commercial & Agricultural Intelligencer (Sydney, New South Wales, Australia) of Wednesday 28^th July 1841 [page 5, column 1]—here, one of his employees is speaking to the farmer:
—Note: Also of interest in this text is “my mouth and throat […] filled with cobwebs”:

“I am well and happy. I work with ease, sleep sweetly, and when I get up in the morning, instead of having, as I used to do, my mouth and throat (to use his own words) filled with cobwebs, as to be constantly spitting cotton wool all the time, my mouth and throat are clear as a whistle.”

1.3-: From the Barton County Democrat (Great Bend, Kansas, USA) of Thursday 23^rd April 1896)—cf., below, quotation 2.3:

“I have been spitting feathers and gun cotton for a month.”

2-: Early occurrences:

These are, in chronological order, the earliest occurrences of the phrase to spit feathers that I have found:

2.1-: From A Negro Quilting Party, or The Salted Whisky, published in the Valley Spirit (Chambersburg, Pennsylvania, USA) of Saturday 20^th July 1850 [page 1, column 2]—reprinted from The Flag of our Union—choaked is a variant of choked:

The sweetened water [was] scarcely tasted, although they were half choaked; they did not mean to satiate their thirst with sweetened water when whisky punches were in prospective. They consequently conquered their thirst with the idea of ‘a good time coming,’ and commenced dancing […].
[…] Next in order came the supper. The young bloods took the damsels by the hand and led them into the refreshment room, where a table burdened with eatables—including hoe cake—awaited them. A long array of tumblers for the whisky were stationed on the table like the wigwams of a tribe of Indians. But where was the whisky? where? echo answers ‘where?’ The hostess came into the room, her face nearly white, and with a quivering voice told the assemblage that her ‘old man’ had forgot the whisky and sugar. It had entirely slipped the observation of the whole family. What was to be done? The dancers had danced and sweat, and were nearly choaked. They already spit feathers, as they termed it. But the whisky they would have and they would restrain their thirst for a time longer.

2.2-: From the column Higglings, by ‘A Hold Hidiot’, published in The Clinton Public (Clinton, Illinois, USA) of Friday 22^nd December 1893 [page 10, column 3]:

Ah! Christmas! Christmas! How welcome art thou to those whose pocket-books have golden linings! They smile now in their pleasant homes. Mince pies and cranberry marmalade are getting ready on the upper shelf, out of little Billy’s reach; oysters are coming into town by the cart-load, in demand for “stuffing;” cider, both crab and Smith, is barreled into market from the country suburbs; and the mouths of the rich are spitting feathers in anticipation of the luxurious treat a-coming.

2.3-: From the Barton County Democrat (Great Bend, Kansas, USA) of Thursday 23^rd April 1896 [page 4, column 2]:

Before the recent soaking rains, a man came down from the divide, north of Barton county, to do some dickering in Hoisington. “Pretty dry up your way,” remarked a store keeper. “Dry? Well rather. […] Why it was dryer than a Kansas republican when he gets off the train at Kansas City union depot. I have been spitting feathers and gun cotton for a month, and instead of sweating the perspiration rattles out of our hides in little chunks of rock salt.”

Perhaps with humorous allusion to its sense ‘to be parched’, the phrase to spit feathers is used in the following text with the exact opposite meaning, since it refers to persons who have been drinking a lot. Here, the noun feathers jocularly refers to the noun cocktail, punningly interpreted as designating the tail of a cock—in the same manner, in this text, the noun leather jocularly refers to the noun cobler (a variant of cobbler), which designates an alcoholic mixed drink.
—This is the text in question, from A Trip to New York. From Our Own Correspondent, dated New York City, Thursday 27^th July 1854, published in the Detroit Daily Free Press (Detroit, Michigan, USA) of Wednesday 2^nd August 1854 [page 2, column 3]:

At half past ten P. M. we perceived six of those eminent livers anxiously watching the amalgamating manipulations of the gentleman behind “the bar.” Mint and brandy were sent to smash with an affectionate lingering of the lips; gin rooster narratives—vulgarly speaking, cock-tails, were pulled out of tumblers, and consumed until some spit feathers; while so many “coblers” were destroyed that the leather market became fluctuating to such a degree, that even the boots on their feet wouldn’t “remain firm,” as the breadstuff reports say.