At the risk of very much being "that guy," it really comes down to what we mean when we use the term republic. In modern usage, the word customarily refers to a regime without a king governed by some kind of elected representative body. In reference to the era of the founding of the United States, this meaning is a bit anachronistic.

The English word "republic" derives from the Latin res publica, itself sometimes a rendering of the Greek politeia. The Latin term directly translates to "public thing," and it is hard to capture the exact classical meaning in English, and the term could mean somewhat different things depending on context. In the early modern period, it became common to use the word "commonwealth," which gets close, but doesn't fully capture the significance of the term's meaning. In its classical usage, a republic did not necessarily refer to a specific type of regime. The word as it has been passed down to us is the word as refracted through the analysis both of Aristotle and of Polybius. They used the term to describe a society that was well-governed by an orderly political system that did not tend too strongly towards monarchy, democracy, or oligarchy. Specifically with reference to Polybius, the term "mixed constitution" shows up quite a bit. Both Aristotle and Polybius argued that the ideal state would be governed under some kind of mixed system. Monarchies could too easily become tyrannies, and democracies very easily became disorderly. A well-governed commonwealth could see to its affairs if it avoided the extremes of any single system, and borrowed beneficial traits from more than one. The republic referred more to a regime that permitted a particular type of social and political order than to the exact constitutional form of the regime itself.

What this meant was that a regime could actually have a monarch and still be considered a republic. Romans did not consider the rise of the principate under Augustus, for example, to have completely ended the republic because the word republic did not mean the same thing for them that it does for us. Elevating particular men to the role of Princeps was a way of restoring republican order when the Senate had manifestly failed to do so. This understanding of the Roman polity persisted for a remarkably long time, and well after Rome's elected institutions had been eclipsed by the Emperors. Even in its Byzantine period, Romans continued to understand a republic by these terms, and you can actually find Frankish sources from the early middle ages describing kings and churchmen ruling over a res publica. The exact shape the regime took was not the fundamental variable; a republic was a well-ordered commonwealth governed by law. Machiavelli was one of the first major writers to employ the more modern usage suggesting the absence of monarchical rule, but both meanings continued to be used alongside each other.

So what exactly does this well-governed society look like? There is no single answer to this question, but with reference to the term as it came to be used in English North America after being refracted through classical usage and the English republican controversies of the 1600s, we can make some generalizations. Following classical usage, a republic was governed under the rule of law and for the sake of the common good. This is not a utilitarian aggregation of interests; society was conceived of as one organic whole, and there could be only one common good for that society. Classical republicanism did not recognize the legitimacy of multiple competing interests in the same society. It would be unjust for the merchant interest, or the banking interest, or any single segment of society to pursue its own goals at the expense of others because that would be detrimental to the common good. This is where a lot of the anti-party rhetoric you ask about comes from. Parties were thought to be bad because they promoted the interests of their supporters over those of the commonwealth. Anglo-American republicanism also has a very strong agrarian streak to it. Farmers were considered more independent because they worked for everything they had, and depended on nobody else. The farmer is uniquely tied to the republic because his livelihood - the land - is intimately tied to the health of the commonwealth by virtue of the fact that it can't be moved. If the republic falls, the free farmer falls with it. In contrast, merchants and those with moneyed wealth can always move somewhere else; their loyalty was said to be to commerce rather than to the republic. When Thomas Jefferson organized political opposition to Washington and Adams in the 1790s, he did so in the name of the common farmer and against the moneyed "monocrats" in the coastal cities. And what does he call his party? The Republican Party, as in, "we are the only ones actually defending the republic. We aren't really a party, we're just standing up for the common good. Now those other guys, those guys are definitely an untoward faction, and should be stopped." American rhetoric about "real America" and "taking our country back" comes from this notion established early on that there was only one common good, and that there was no legitimate disagreement about what that common good was. Either you were for the republic, or for some noxious faction

When we get to the American founding era, this is more or less what a republic is taken to mean. The men who wrote the founding documents either had a classical education, or were conversant enough in that world to have some sense of what these words meant and how they were used. Even by this point, the word "republic" does not exclusively refer to a regime without a king. Here's John Adams in one of his Novanglus letters dated to 1763:

"[T]he British constitution is much more like a republic than an empire. They define a republic to be a government of laws, and not of men. If this definition be just, the British constitution is nothing more nor less than a republic, in which the king is first magistrate. This office being hereditary, and being possessed of such ample and splendid prerogatives, is no objection to the government’s being a republic, as long as it is bound by fixed laws, which the people have a voice in making, and a right to defend. An empire is a despotism, and an emperor a despot, bound by no law or limitation but his own will; it is a stretch of tyranny beyond absolute monarchy."

Adams wrote this at a time when Americans were still trying to defend the British constitution as they understood it. Note that on his terms, Britain is a republic even though it has a king. The push for independence came when those responsible for making those kinds of decisions came to believe that authorities in London were replacing this mixed constitution with a species of tyranny that governed only in the interest of the few. This fear of tyranny and fear of bad government is the root of the anti-party rhetoric expressed by George Washington and others.

As to whether we are a republic now, that is a question of personal judgment that heavily depends on how you think that term is defined, and what you think is important.

Readings

Paul Rahe, Republics Ancient and Modern (3 vols)

Bernard Bailyn, The Ideological Origins of the American Revolution

Gordon Wood, The Creation of the American Republic, 1776 - 1787

Lance Banning, The Jeffersonian Persuasion: Evolution of a Party Ideology

Drew McCoy, The Elusive Republic: Political Economy in Jeffersonian America

Anthony Kaldellis, The Byzantine Republic: People and Power in New Rome

Elkins and McKitrick, The Age of Federalism - The Early American Republic: 1788 - 1800

Andrew Lintott, The Constitution of the Roman Republic

J. H. Burns, The Cambridge History of Medieval Political Thought: c.350 - c.1450