Filibuster was used from 1850 to refer to American adventurers in private military expeditions against Cuba and other countries to the south of the United States. And by December 1859 it was already in use with reference to political manoeuvring and delaying tactics in the House of Representatives.
Etymoline misses some of the early uses, but otherwise, as far as I can tell, their article on the topic is rather good. They say the noun appears in
American English, from 1851 in reference to lawless military adventurers from the U.S. who tried to overthrow Central American governments. The major expeditions were those of Narciso Lopez of New Orleans against Cuba (1850-51) and by William Walker of California against the Mexican state of Sonora (1853-54) and against Nicaragua (1855-58).
There's a slightly earlier reference, in the Baton-Rouge Gazette, August 10, 1850, in reference to the Cuba expedition (my boldface in all quotes):
Nor will reference ever be made again to the Crescent’s record that President Taylor’s performance of duty, in attempting to prevent the Filibusters from taking the Island of Cuba was certainly extremely distasteful.
Earlier in the same article filibuster appears twice, along with Filibustrian doctrine, and it appears to be in connection with expansion of Texan territory into Mexico. By early 1851 the word had gained wide currency and there was interest in its origin already, as stated in The Daily Crescent, Baton Rouge, La., January 14, 1851:
“The Filibuster.”―This word, which has been figuring in every press of the land, since the Quixotic expedition of Cuba, afforded Mr. Benjamin a fair field for some facetious ridicule. At the close of his remarks, Mr. Benjamin introduced this upstart, and raised to the jury that it is a corruption of the Spanish word flibustero, which is but another appellation for a buccaneer. Without much straining―we do not wish to say with the fact on our side―we could trace it to a better and more germane source. Thus the Yankee begot filibuster out of the Spanish flisbustero, which was begotten out of the French flibustier, which was corrupted out of the English free booter.
For the verb in the ‘insurrectionist’ sense Etymoline quotes from Harper's New Monthly Magazine, January 1853:
FILIBUSTERING is a term lately imported from the Spanish, yet destined, it would seem, to occupy an important place in our vocabulary. In its etymological import it is nearly synonymous with piracy. It is commonly employed, however, to denote an idea peculiar to the modern progress, and which may be defined as the right and practice of private war, or the claim of individuals to engage in foreign hostilities aside from, and even in opposition to the government with which they are in political membership. [“Harper's New Monthly Magazine,” January 1853]
And Etymoline on the ‘legislative sense:’
The noun in the legislative sense is not in Bartlett (1859) and seems not to have been in use in U.S. legislative writing before 1865 (filibustering in this sense is from 1861). Probably the extension in sense is because obstructionist legislators “pirated” debate or overthrew the usual order of authority. Originally of the senator who led it; the maneuver itself so called by 1893. Not technically restricted to U.S. Senate, but that's where the strategy works best.
It goes on to say:
[The 1853 use of filibustering by U.S. Rep. Albert G. Brown of Mississippi reported in the "Congressional Globe" and cited in the OED does not refer to legislative obstruction, merely to national policy toward Cuba.]
U.S. Rep. Brown’s 1853 use of filibustering (Congressional Globe, found by lly; see lly’s answer) is however relevant because it refers to someone, U.S. Rep. Venable, giving a speech in the House, not attacking another country:
But my friend from North Carolina has advanced another argument which is yet more extraordinary, extraordinary I mean as coming from him. When I saw my friend standing on the other side of the House filibustering, as I thought, against the United States, surrounded, as he was, by admiring Whigs, I did not know what to think. It seemed to me he had taken formal leave of his old State-Rights friends, and gone over to the Whigs.
In what follows Mr. Brown does not accuse Mr. Venable of delaying tactics; he just criticises the content of the speech. So he appears to use filibustering figuratively, meaning that Mr. Venable is working against United States’ interests.
But by December 1859 verb and noun were already being used in connection with political manoeuvring around the election for the Speaker of the House of Representatives, often referred to as “organization.” A little background is in order. There was no majority in the House, as the Republicans had 109 seats, the Democrats 101, and the Know Nothings or American Party 27. Tensions were running high on the issue of slavery, and had recently been inflamed by the publication of the anti-slavery The Impending Crisis of the South by Hinton Rowan Helper (Wikipedia). Pro-slavery Democrats were furious at the book, whereas Republicans used it in their campaign. Republican John Sherman was a leading candidate for Speaker, but Democrats were dead against him or any of the Republican that had expressed support for Helper’s ideas. (John Sherman in Wikipedia). The verb filibuster keeps appearing almost daily in the American press that December. A few examples:
The signs are favourable for a Republican organization. Several Democrats are absent, and Clarke, of Mo., has introduced a fillibustering-resolution that no man who recommended Helpers’s “Crisis at the South” is fit to be Speaker. This is only to gain time. It is a poor dodge which wont amount to much.
(Holmes County Republican, Millersbourg, Holmes County, Oh., December 8, 1859.)
So you see the slavery-democracy is factious as well as fractious. Their motto is “rule or ruin.”―Finding themselves in a minority they filibustered―not believing in the majority principle […] The Democracy cannot succeed without the aid of the South Americans, and with the hope of forcing them to come to the support of Bocock they are resorting to all manner of factious proceedings to stave off action and prevent the election of a Speaker.
(Delaware Gazette, Delaware, Oh., December 9, 1859.)
This is all the Republicans can count upon, and it falls short of a majority of the whole House by just ono vote.' This being the relative strength of parties, the great object with those desirous of an early organization seems to be to adopt the plurality rule. But even in the event of this being done, the fire-eaters and doughfaces will doubtless filibuster to secure farther delay.
(The Press and Tribune, Chicago, Ill., December 14, 1859.)
Although more than a week has elapsed since the meeting of Congress, there has been no choice of a Speaker by the members of the House of representative". Up to Tuesday night but three ballots had been taken―almost the whole of the week having been consumed by the Administration party in filibustering and storming about Helper's book.
(The Jeffersonian, Stroudsburg, Pa., December 15, 1859.)
After the questions of privilege are disposed of some Republican member jumps up and sings out, “I move that we proceed to ballot for Speaker.” Cries of no, no, from the Democratic side and yes, yes, from the Republicans. A Babel o tongues ensues. The Democrats begin to filibuster; to move that a vote be taken on Clark’s Helper book resolution or move to adjourn, and call the yeas and nays, or move a call to the House and the yeas and nays on that
(The Press and Tribune, Chicago, Ill., December 26, 1859.)
And in this article in The New York Herald, December 17, 1859, we have the people filibustering referred to as filibusters:
I hear a report that the Americans will vote for Boteler until he withdraws, when they will concentrate on Ehteridge, and notify the republicans that they cannot have the plurality rule and must unite on Mr. Etheridge. Such a course will only put off the organization to an interminable period; and the responsibility will rest with the factious filibusters.