"I feel myself unhappy" vs "I feel unhappy"
The wording "I feel unhappy" is commonplace in English today. "I feel myself unhappy" is, by contrast, rare and sounds odd to many native English speakers. This was not always the case, however. In the late 1700s, "I feel myself unhappy" had considerable support in published English, and "I feel unhappy" was not readily to be found.
'I feel myself unhappy' in the late 1700s
Several occurrences of "I feel myself unhappy" from the 1770s and 1780s involve luminaries of the American Revolution and subsequent American Republic. From a letter by Abigail Adams to John Adams dated October 21, 1775, in The Quotable Abigail Adams (2009):
Expectation has so long and so often been combatted by dissappointment that I feel myself unhappy, my Spirits which were naturally cheerfull are depressed and the enjoyments of life are growing very insipid to me.
From a letter by George Washington to Governor Jonathan Trumbull (May 23, 1777), in The Writings of George Washington from the Original Manuscript Sources 1745–1799, volume 8 (1933):
All the [military] Stores are coming to Springfield, where they will be deposited in the Public Magazine, except such as will be immediately wanted for the Army, and I feel myself extremely unhappy, in not having it in my power to consent, that a part should be appropriated as you request.
From a letter from Abigail Adams to James Warren (ca. June 7, 1778) in Adams Family Correspondence, volume 1 (1963):
Tis now four months since the Boston saild in all which time we have never heard a word from our Friend's. Our Enemies tell us that She is taken and carried into Plimouth. I know not what to think, but I feel myself unhappy and the more so I think for hearing a day or two ago, that a French vessell had arrived at Portsmouth with News that the Boston was safe arrived in France but it proves that tho she left France near 8 weeks after the Boston saild from here that She heard nothing of her.
From a letter by Colonel Van Schaick to New York governor George Clinton (May 27, 1780), reprinted in Documents of the Senate of the State of New York, volume 9 (1901):
Sir; I am favoured with your Excellency's letter of yesterday. I feel myself unhappy that I have not been able to pursue the Enemy previous to the receipt of your letter. I expected when I left Albany to have been able to Collect a Considerable force at this place, but on my Arrival, which was on Wednesday the 24th, I could not muster more than One Hundred & Eighty men; Thursday 250, & on Friday 500; while I was Stimulating the militia to Join me, I received repeated accounts that Sir John continued his Camp at a Settlement called Mayfield; my Scouts which I kept constantly out returned on Thursday & brought me an account that Sir John had decamped from Mayfield, on Tuesday, ...
From a letter by George Washington to Meshech Weare (August 21, 1781) in Collections of the New-Hampshire Historical Society, volume 2 (1827):
Sir, — I feel myself unhappy in being obliged to inform you that the circumstances in which I find myself at this late period, have induced me to make an alteration of the main object which was at first adopted, and has hitherto been held in view, for the operation of this campaign.
From a letter by Governor George Clinton of New York to Nathaniel Gorham, President of the Congress of the Confederation (August 16, 1786), in Journals of Congress, 1786 (1823):
A copy of their [the New York legislature's] act passed on the occasion, I have had the honour of laying before Congress through the delegates of this state, and your excellency [Gorham] will readily perceive it is not my province to determine how far it conforms to the recommendation in question. I cannot conclude without adding, that I feel myself unhappy to be formally called on by Congress, in an instance in which I cannot yield a compliance without breaking through one of those checks which the wisdom of our constitution has provided against the abuse of office, and which, I am persuaded, Congress will approve the sentiment, when I declare, I find myself bound, as well by inclination as by duty, to preserve.
And from remarks by Representative Michael Stone of Maryland in the House of Representatives on September 3, 1789, reprinted in The Debates and Proceedings of the Congress of the United States, volume 1 (1834):
The main question being now before the committee,
Mr. STONE proceeded. I feel myself unhappy to be obliged to address gentlemen, who are not disposed to attend to any thing I may say; but as gentlemen have chosen this time for discussing the subject, they will no think it improper in me to persist in detailing my ideas.
'I feel myself unhappy' in Great Britain
Lest it appear that "I feel myself unhappy" was exclusively a colonial barbarism, I note a couple of early instances from the British Isles. From Irish playwright Arthur Murphy, All in the Wrong (1761) in The British Drama, parts 1–2 (1804):
Belinda {Without looking at him [Beverley].} You have kept me on the rack this whole day, and can you wonder that I feel myself unhappy?
And from a letter by London pastor Andrew Fuller to the church at Kettering (received October 21, 1781) in The Work of Faith, the Labour of Love, and the Patience of Hope Illustrated in the Life and Death of the Reverend Andrew Fuller, Late Pastor of the Baptist Church at Kettering (1816):
I am at this time a compound of feelings. I feel, dear brethren, I feel painfully for you. I am distressed that a church whose troubles were many before, should have them increased through me. I feel myself unhappy lest my worthy brethren and fathers in the ministry should think themselves slighted, of which there is nothing that I am less conscious: and should they on this account slight me it will very much grieve me, but that I cannot help it.
The phrasing is also used, somewhat later, in Thomas Carlyle, Wotton Reinfred (written 1826–1827), published in The Last Words of Thomas Carlyle (1892):
Jaspar was not with us but in England at a boarding-school; one grave old woman was our only servant. Yet this solitude was not lonesome to me, nor with all my little griefs did I feel myself unhappy. What wealth is in childhood, how that morning sun makes a very desert beautiful! One has yet no consciousness of self, one is a thought, an action, not a thinker or an actor.
Early instances of 'I feel unhappy'
A Google Books search for "I feel unhappy" returns matches beginning at the very end of the 1700s. From Isaac D'Israeli, The Lovers, or the Origin of the Fine Arts (1799):
It is in winter I feel unhappy: the cave our fathers hewed is so round, or so square, one sees the termination of every thing; nothing is left to the imagination. It is not thus in Nature; she never imprisons the eye; all her lines are waved, and varied, and enchanted.
From Margaret Cullen, Home: A Novel, volume 1 (1802) [combined snippets]:
"You have no occasion to request ; you have only to state the facts I shall mention to you, and your father himself will see the propriety of complying with your brother's desire. You can have no objection to this.——Why are you silent, Constantia?"
"Because I feel unhappy at the idea of speaking to my father on such a subject."
From Regina Roche, The Discarded Son; Or, Haunt of the Banditti: A Tale, volume 1 (1807):
But that calmness and cheerfulness which his arguments with himself sometimes failed of producing, the consideration of his children's now smiling prospects, in a moment restored him to ; and 'how can I feel unhappy?' was a question he still asked himself whenever they recurred to his recollection, 'how complain of the unkindness of fortune, when in their assured happiness I behold my fondest wishes accomplished?'.
All three of these authors are from the British Isles—D'Israeli from England, Roche from Ireland, and Cullen from Scotland.
Conclusions
My research doesn't establish that "I feel myself unhappy" antedates "I feel unhappy," but it does strongly suggest that in the late 1700s and early 1800s English speakers—especially in America— did not consider "I feel myself unhappy" to be improper, redundant, or peculiar.
Today, as the question by user79773 and the answers by Harrison and SteveRacer indicate, the vastly more common wording is "I feel unhappy"—and the wording "I feel myself unhappy" seems awkwardly overstated. Indeed, if we may judge by Google Books search, "I feel myself unhappy" has been out of use in newly published writing for at least a century. But historically both wordings have appeared in English, and the meaning of the two phrases is virtually indistinguishable.