We do have Old Latin. Latin changed just like any language, but there are processes which can crystalize a language and retard its change. This happened to Latin to some extent (though not completely) because it was a prestige language and the language of Imperial administration. It was also a historical literary language, especially Golden Age and Silver Age Latin, so when a modern student learns Latin, they tend to learn Golden Age Latin, the Latin of Cicero and Caesar and Vergil and Ovid and Livy of the first century BCE. It is for similar reasons that modern students typically learn "Attic" (Athenian) Greek of the Classical period: there was a flourishing of writing in this dialect (Thucydides; Plato; Aristotle; virtually all Greek tragedy and comedy, e.g.).

There are very few literary texts of Old Latin, especially complete ones, but we have substantial fragments from Archaic Latin authors: e.g., Livius Andronicus, who wrote a Roman adaptation of the Odyssey; Naevius, who wrote tragedies and comedies; and Ennius, who wrote an earlier version of the Aeneas tale. We also have a nice body of inscriptional evidence which show the quirks and alternate grammatical constructions of early Latin, some going back into the 5th century BCE or even earlier.

Many of the most substantial literary fragments are collected in Erasmo, Archaic Latin (Focus, 2001)(for specialists only, as there are no translations and the Latin itself tends to be difficult).

On the other end, Latin of course changed beyond the Golden Age period. In the first century, we speak of "Silver Age" Latin, when some of the grammar and spellings and such began to change somewhat under the Empire. Some notable Silver Age authors are: Seneca, Columella (cookbooks), both Plinys, Tacitus, and Marcus Aurelius.

And into the Late Imperial period of Late Antiquity, the language continued to change. After the end of Roman rule in the west, the fragmented areas of Latin speakers developed their own vulgate brands of Latin which became the early versions of the Romance Languages.

In short: we do have multiple versions.

Latin evolved a lot over the centuries. It was still a live language introducing new terms and uses clear into the 15th, or so, Century. Medieval Latin was a lot different than classical Latin. We just don't teach it in schools, unless you are aiming at a very specific specialization.

Consider this. Latin is a taught language. It isn't really anyone's mother tongue. So, the tendency is to pick a specific baseline for the language. Latin was the international language for communication clear until the late 17th, early 18th Centuries. It was still used for scientific and math publications. In that sphere, Classical Latin was only adopted as the version of Latin to be taught during the Renaissance. Prior to that, it was just the version of Latin that was was still live and it was incredibly different from Classical Latin.

Even in classical Latin, you can see the language changing when phrases like "hoc dies" evolved into "hodie" as well as a lot of other changes.

But, consider the day-to-day language spoken by people. These were not locked down to specific a language that was specifically taught to non-speakers. They evolved. Some slowly and some radically based upon the trajectories of their regions after the fall of the Western Roman Empire. During the time of what we consider Classical Latin, there was also a Vulgar Latin and a dozen different flavors of Latin. Classical Latin was the specific version that ancient Roman grammarians wrote volumes of books about. If you were to pick a version to teach, what we settled on has a lot going for it over the other versions just from the shear amount of existing documentation from native users on the framework of the language.