In American English, the slang expression blue funk denotes a state of depression or despair.

This phrase occurs, for example, in a problem put to the U.S. author and journalist Carolyn Hax (born 1966) by a person signing themself ‘In a Funk’, in the advice column Ask Carolyn, published in many newspapers on Sunday 20^th March 2022—for example in the Argus Leader (Sioux Falls, South Dakota, USA):

Hello, Carolyn: Our adult daughter, 28, doesn’t call us often, but often when she does, she is in a blue funk and sees every aspect of her life as negative.

The earliest occurrence of blue funk used in the sense of a state of depression or despair that the Oxford English Dictionary (OED – online edition, March 2023) has recorded is from Blue Funk in the Fourth Decade, by the U.S. physician and medical columnist William Brady (1880-1972), published in many newspapers on Friday 14^th February 1958—for example in the Asbury Park Evening Press (Asbury Park, New Jersey, USA):

A Michigan reader reports that last winter, after 46 years of generally excellent health, she found herself in a long period of low energy and “blue funk.”
The term “funk” signifies frightened, to shrink from or to flinch or, as morons say, to “chicken.” But it seems evident that “blue funk” in this instance means merely low in spirit, gloomy, melancholy, pensive, moody, hypochondriac or, to coin a word, anhedonic.

According to the OED, in blue funk used in the sense of a state of depression or despair, the noun funk (of Scottish origin) denotes a state or fit of gloom, bad temper, depression, irritation, etc.

Therefore, if the OED is to be believed, there would be two distinct expressions:
a) the American-English expression blue funk;
b) the British-English expression blue funk, which denotes a state of extreme nervousness or dread, and in which the noun funk (which originated in the slang of the University of Oxford, Oxfordshire, England) denotes a state of extreme nervousness or dread.—Cf. ‘blue funk’ (British and Irish usage).

However, the earliest occurrences that I have found of blue funk used in the sense of a state of depression or despair indicate that this American-English acceptation simply is a shift in meaning of the British-English expression, which was borrowed into American English in the late 19^th century.

In fact, in early American-English usage of blue funk, this expression was used in the British-English sense of a state of extreme nervousness or dread, and was frequently referred to as being of British-English origin. The following, for example, is from The People at the Polls, published in the Springfield Daily Republican (Springfield, Massachusetts, USA) of Tuesday 3^rd November 1874:

Seriousness is one thing and panic another. The average American voter is a being of weaker nerves and greater gullibility than he has been generally supposed, if he allows himself to be thrown into a “blue funk,” as our English cousins phrase it, by the alarmist talk with which interested politicians and their newspaper attachments are trying to scare him into a surrender of his private judgement and his inalienable right of thinking and choosing for himself.

The earliest occurrences that I have found of the expression blue funk in the sense of a state of depression or despair are as follows, in chronological order:

1-: From Manuscript Returned, Not Rejected, a story told by a writer, published in the New-York Tribune (New York City, New York, USA) of Sunday 10^th December 1893:
—This text clearly illustrates the shift in meaning of the British-English expression blue funk:

“I sent off [the manuscript] in a sort of ecstatic dream. I had not the slightest doubt about it. I knew that it would be several days before I got an answer—the usual days—but that delay was not waiting; it was only ‘clinching the thing.’ And then it came. I was in my library when the maid brought it to me. It was returned. There it was—the big, fat envelope. I knew what the note which accompanied it would say, and I would not read it. I did not even open the envelope. I laid it aside, and I must confess that just then I lost faith in a certain editor, a warm friend of mine. If he did not recognize the merit of that bit of work he could not be the able editor some people, including myself, had believed him to be. I have never felt worse over a rejected manuscript than I felt over that. I got into a depth of gloom that was appalling. I just sat there in my library in what Englishmen call a blue funk.”

2-: From The Sun (New York City, New York, USA) of Monday 24^th September 1894:

Why Art Thou Cast Down?

Why is the Hon. William M. Singerly cast down because the Hon. William McAleer has not been renominated for Congress? […] Surely he, the old soldier in the truceless war against protection, should have rejoiced and not wept when a deserter like McAleer was laid by the heels. Surely Col. Singerly, instead of howling “Ichabod,” and tearing off the buttons of his uniform, should have shouted “So perish all traitors!” and honed his trenchant blade upon his cuffs with a deadlier resolution.
But every great captain has had his hour of weakness and despair, his moments of blue funk.

3[?}-: From Our New York Letter, by Dexter Marshall, published in The Daily Advocate (Stamford, Connecticut, USA) of Tuesday 7^th June 1898—however, here, blue funk may refer to fear:

New York, June 7.—[Special.]—The admirers of Mr. James T. Sloane, professionally known as Tod Sloane, are not yet over the blue funk into which they were thrown the other day when, mounted on Lehman at Gravesend, he “rode like an amateur,” making a “weird and fearful exhibition” which lost the race and convinced every one that lapses may be the occasional lot even of an idolized popular jockey.

4-: From a letter from the U.S. journalist Byron Rufus Newton (1861-1938), of the Associated Press, dated Port Antonio, Sunday 17^th July 1898, published in the Buffalo Evening News (Buffalo, New York, USA) of Thursday 28^th July 1898—in the following, Byron Newton described U.S. soldiers going insane at Siboney, near Santiago de Cuba, Cuba:

There is something interestingly gruesome and grim about the condition of the men one sees about hospitals and camp. The terribly depressing atmosphere of death and suffering all about them seems to bring on a state of semi-coma. We call it “blue funk.” I have seen men ordinarily light-hearted and full of life sit by the hour about the yellow fever camp at Siboney, seemingly interested in nothing except the work of death. It is a strange condition. Men become sullen and melancholy, apparently incapable of being aroused by anything except those things which are, under ordinary conditions, too hideous to be endured.