They were absolutely aware. For as long as people have been writing, they've been writing about language. And for as long as there's been writing about language, people have been noticing it change and often commenting.

You have to understand that throughout most of history there has been no such thing as a standard form of the language, at least not one that went thru unchanging across generations. There were people that were proposing standards of different registers of the language, Pāṇini being one of the earliest and most well known. But throughout time and especially in the regions/times you've brought up, people were well aware of the regional dialectal variations and the ways in which their languages were changing over time. In Greece alone there were a good number of regional dialects and it was a combination of two of those that eventually developed into the modern language. And people were aware of what was happening as it was happening.

Likewise with Latin, the form of the language you're going to see (even as a written prestige form) is going to differ between the Republic and the end of the Empire. Same for Classical Chinese, same for Sumerian (though records here are somewhat sparse, Sumerian is a great example of language change because a lot of what we know about it we know from Akkadian and the Akkadianisation of the people's language).

As you said:

The modern study of linguistics supposedly started when the British noticed similarities between Sanskrit and Classical Greek.

This is a fairly Euro-centric view of linguistics, and actually many would argue the modern study of linguistics happened later than that even. In China, while there were people who were influenced by the goings-on of people like Grimm & co., they'd already been doing their own sort of modern linguistics since at least the 6th century CE, and an earlier form in at least the 3rd century BCE.

At any rate, in everywhere where there's been ancient philosophy, the philosophy of language has been a part, to include usage and changes in usage.

The start of modern linguistics, however you want to define it, doesn't really mean it's when we first became aware of language variation and first became interested in figuring it all out. It's just an arbitrary jumping-off point where something changed in how we do it which, to some extent, is how we're still doing it. For the same reason that you can look at the period you mentioned as the start, others might call 1968 the birth of modern linguistics.

Yes. The Romans saw the similarities between Greek and Latin, and believed Latin to be a dialect of Greek. I will direct you to this article for more on the subject.

There are some lost works of Varro that would be very enlightening on this, namely the De Origine Linguae Latinae, On The Origin of the Latin Language.

I did a quick search in the Packard Humanities Institute tool for instances of latine near graece in the corpus there, and got some cool results, but I don't know how much Latin or Greek you can read. Just picking a few little instances out, we can see Cicero noting the similarity of the Greek στάσις and Latin status, Tiro comparing Greek ὕες to Latin sues, (though I'd like to have seen something more said about h to s shift which is so common from Greek to Latin), Aulus Gellius comments on the Greek φώρ and Latin fur, etc.

here is a fascinating letter by Cicero which deals with a different kind of language change. He says, among other things, that penis used to mean tail, but know it has changed to be an obscene word.

Since you say you're interested in specific mentions of usage changing, you might want to take a look at the Appendix Probi. It's a list of about 270 commonly (at least, apparently commonly to whoever compiled the list) misspelled words.

The modern study of linguistics supposedly started when the British noticed similarities between Sanskrit and Classical Greek.

"Modern linguistics" has several commonly given starting points from two different founding fathers. What you have to understand is that while explicitly academic work on language was done since William Jones, it was focused on what we would now call historical linguistics. Jones set in motion for others to start seriously considering how languages may have been descended from each other, and that was the start of philology. The first founder of linguistics Ferdinand de Saussure, was the first to think about how language as a system works. That statement may strike you as odd considering the various grammatical treatments of specific languages that have been done over time, with Panini's study of Sanscrit being the oldest known, but Saussure was interested in language as a system, not specific languages with their specific grammars. He was a pioneer who made two key distinctions that are still hallmarks of the post-philology study of language today, that of the synchrony-diachrony distinction and the langue-parole distinction (which is essentially the same as Chomsky's competence-performance distinction). The second commonly attributed father of linguistics is Noam Chomsky. This is a title bestowed on him for the start of generative linguistics, which inspired and pioneered a lot of work in formal language representations (and is incidentally the reason virtual assistants like iOS's siri understand you via Head-driven phrase structure grammar parsers).