The short answer is "yes," at least some Romans in Caesar's day understood that their language changed over time. A great example is the author and scholar Varro, who was a contemporary of Caesar's (and indeed was pardoned by him during the civil wars). Varro wrote voluminously on the Latin language and its history. That includes his 25 book "On the Latin Language" which survives in part, as well as several other works on Latin Antiquities and other subjects touching on language which are lost (or survive only in fragments quoted in other works).

His surviving works include sections on the subject of etymology and the origins of usage, which display a sophisticated understanding of how Latin changed over time. For example, the following paragraph appears in the fifth book of "On the Latin Language":

These relations [the relationships between words and their meanings] are often rather obscure for the following reasons: Not every word that has been applied, still exists, because lapse of time has blotted out some. Not every word that is in use, has been applied without inaccuracy of some kind, nor does every word which has been applied correctly remain as it originally was; for many words are disguised by change of the letters. There are some whose origin is not from native words of our own language. Many words indicate one thing now, but formerly meant something else, as is the case with hostis, 'enemy': for in olden times by this word they meant a foreigner from a country independent of Roman laws, but now they give the name to him whom they then called perduellis, 'enemy.'

So just in this little paragraph you can see an awareness that word meanings and spellings of words can change over time, that words are borrowed from other languages, that words die out, etc. His books go on at much greater length on these subjects, the quoted example is laying the groundwork for the specifics that follow.

Certainly Varro wasn't infallible, and some of his etymologies are now viewed as erroneous by Latin scholars, but he's an excellent contemporary source and you'll still find cites to him in modern Latin dictionaries for etymologies or for certain Latin words that meant different things at different times.