Are you asking if the actual sources survive? Like the actual letters that Cicero wrote himself? Or Thucydides' actual manuscript that he wrote? If that's the question, then the answer is largely no.

The only real primary source documents (for long-form writing) are inscriptions on stone. And there are quite a few of those. And there are some rare exceptions, such as the Herculaneum papyri which were buried by Vesuvius in 79 AD. These are just now being read using advanced imaging techniques (all very, very dull philosophy).

After that, everything survives because of ancient libraries and the medieval church. Anything worth preserving in the eyes of the church, which included histories, satires (which the church deemed worthy because they moralized poor character), and examples of great rhetoric, were copied and recopied throughout the middle ages. The only great libraries were ecclesiastical libraries. And it's because of them that these things survived into the era of the printing press.

There are still some curiosities that got through though. For example, the poems of Catullus (a great early Roman poet) got pretty obscene (check out Catullus 16, for example). So things like that survive as a kind of medieval dirty book that a monk would copy in his cell at night by candlelight and hide.

And we can trace additions and subtractions from the texts based on their transmission. Manuscripts for one library were copied from the manuscripts stored at another library. So there's a manuscript tradition for every text. So there will be a family of manuscripts that have a peculiar misspelling on the first page, and an added paragraph that isn't in any of the other manuscripts. Those are the 'A' version of the text. Another group is shorter, missing some text in the middle, and doesn't have the misspelling. Those are the 'B' version. In that way, texts can be traced through time to try to get to the 'real' version of the original.

Checking manuscripts against one another (known as collating) shows the complete medieval history of a text. Library A's version was copied by libraries B, C, and D. But Library E's version was copied by libraries F, G, and H. It creates a family tree. And usually there are several versions floating around. But in a few instances, like Catullus, it's clear that every single later copy was derived from one source. That means that that was the only copy in existence for a time. Had there been a fire in that one library, Catullus would've been gone. The plays of Aristophanes were in the same situation at one time.

And if you go back even before that, there were classical scholars that would read texts and amend as they saw fit. That was the main job of an ancient librarian. The librarian at the great library at Alexandria would go through and write commentary in the margins about potential late additions, historical events that they learned of differently, or lines that should be added. Those scholars are known as scholiasts, and are studied in their own right.

As far as 'true' goes, any historian will tell you there's no such thing as truth. Only consensus. But ancient historians will be checked against other historians, against the archaeological record, against surviving inscriptions, and against what we know to be possible for the time period.

If you're interested in the transmission of ancient texts to the present day, N.G. Wilson's Scribes and Scholars: A Guide to the Transmission of Greek and Latin Literature is an excellent read.

We have a lot of really cool ways of knowing about those cultures! For a lot of very ancient sources, especially in Greece, we can look at urns and other ceramic ware with writing and illustrations on them. But even better, and more relevant to your question, we have some very cool documents, written in ancient Greek, some of which are contemporary to the author! It's impossible to tell whether the person credited actually wrote a given manuscript, but if you search [insert author or work here] manuscript, you can find a lot of images of a lot of really cool stuff. Ancient Greece had its own historians too, like Herodotus, and while their actual versions of events are probably biased and have been criticized for their inaccuracies, they still give us a lot of insight on how those individuals thought, and a pretty good idea of what was generally going on. The number of troops at a battle might not be correct, but the battle probably happened.

For Roman authors, we have even better sources. One of the most famous Roman authors is Marcus Tullius Cicero. He got really popular, so we have a whole lot of very old copies of what he wrote, and we think they're pretty accurate because they're fairly consistent. Plus, with stuff like poetry, people get choosy about getting the wording right.

So do we have primary source documents? Yes. Are very many of them written during the time the author was alive? Probably not. But we certainly have some, and we keep finding more.