There isn't a relative lack of written sources generally speaking for the so-called Dark Ages (which I will here be construing narrowly as ~550-800), least of all by way of comparison with the Classical world, nor is this really what is meant by "dark" in this sense of the term. Certainly with the decline of the Roman Empire in the west there is a decline in writing in some areas. To take the common example, little writing survives from Britain between the death of Gildas in ~570 and Bede, writing from around the turn of the 8th century. But the same is true about periods before the so-called Dark Ages. As Walter Goffart notes, we have considerably more in the way of surviving Latin histories from between 550 and 800 than we do for the comparable span of the second and third centuries between Suetonius and Aurelius Victor.

Indeed, the corpus of Classical Latin literature from its beginning to 200 CE is really very small, with estimates based on the PHI corpus putting it around 7.5 million words. By way of comparison, the collected works of Augustine of Hippo alone are in the range of 5 million words and we can use this for a very rough and ready quantitative analysis. Extrapolating from the fact that Augustine's collected works are contained in 15 volumes of the Patrologia Latina, we can note that the same number of volumes cover the period from Gregory of Tours (d. 594) to the last volume before Charlemagne (d. 814). Obviously, this is a very sketchy comparison, since the PL is not consistently chronological, nor is it comprehensive, but it does highlight that in terms of scale of surviving writing, the "Dark Ages" (narrowly construed) produced at minimum roughly the same quantity of surviving Latin writing as everything before 200 CE combined. Nor should this surprise us since typically speaking the survival of writings is largely a function of time, since in almost every instance it is a direct result of their being continually copied by subsequent generations of scribes (at least until the advent of printing, though the dynamic remains largely the same). Indeed, the overwhelming majority of those 7.5 million words of Classical Latin survive through medieval manuscripts alone, with a scant few authors (like Virgil) coming down to us in a few Late Antique manuscripts from the early days of the codex.

But as I alluded to above, this isn't really what is meant by "dark" in this context. I've written at length about this before, most notably I've discussed this specific meaning of "dark" as "lack of sources" in conversation with /u/epicyclorama in this thread and much more generally about the origins of the concept of the "Dark Ages" in this thread). As it is relevant to the discussion here, the usage comes broadly from the Latin term "obscurus", meaning dim or difficult to see, and refers more generally to a lack of knowledge about a period. The classical usage of the saeculum obscurum is by Caesar Baronius referring to the lack of sources for the 10th century papacy. But the early usage of this notion among the humanists, which retains some force still today, is that by comparison with the polished Latin of well-known golden and silver era historians like Livy or Tacitus the medieval sources were not so well known, being especially distributed through a range of sometimes obscure chronicles and annals, and the more holistic works of historiography from the Middle Ages were very much not considered paragons of prose styling. Indeed, this period before the emergence of diglossia between Latin and Romance around the turn of the 9th century is generally the absolute low point in this story. (For a good old-fashioned overview of the quality and quantity of literary Latin production from Antiquity to the Modern era, see this wonderful graph by Walter Berschin. N.b. the top graph is stylistic quality and the bottom is quantity of surviving writing.)

So while there is likely some truth to the suggestion that the scale of writing in the Western Empire declined notably perhaps already from the crisis of the third century but certainly after the end of the Roman education system around the sixth century, this has little principled relationship with the survival of sources.

Well, I’m not sure if you would say that written sources from the era is scarce. The first thing that is worthwhile to note is that much of the literary heritage of the so-called classical world, especially Latin literature, survived to the present era because of copyists and others preserving texts. It took some time for these texts to be “rediscovered,” but the monastic communities of early medieval Europe produced many manuscripts from this era.

Another thing to note at the outset is that it is plausible that text and manuscript production decreased as Western Europe entered the early Middle Ages, that trend is in part technological. As the mediterranean exchange network predicated on the Roman Empire collapsed, papyrus stopped being accessible. So Western Europeans began to write on parchment, which is a product made from animal skin. This meant that it was much more costly to record things. However, there are advantages. First, parchment is able to weather the degradations of age better than most papyrus from the era. Additionally, parchment can be scraped, which means that it is reusable to a certain extent. Additionally, we can use x ray technology to read these scraped pages. Finally, aiding preservation, the sturdier character of parchment allowed for the binding of these texts into codices rather than scrolls. Thus, while there is ample reason for the decrease in texts, the conditions for long-term preservation exist.

And the truth is, we have many, many texts from this era. We have law codes from many of the Germanic groups that established themselves as rulers over the former Roman Empire. Among the most influential and celebrated is the Burgundian Law Code, which dates from the fifth/sixth century, though our earliest manuscript is 9th century. We also have the Lombard code. We also have works of history that range in complexity. Jordanes wrote a history of the Goths in the sixth century. Also in the sixth century, Gregory of Tours wrote a history of the Franks, which Walter Goffart suggested used complexity as a rhetorical device to make comment on the state of the world.

Finally, you have the wholly new production of monastic texts in this era. The early sixth century rule of St Benedict was a landmark monastic text. Of course Bede’s writings in the seventh century represent a high water mark in style and complexity of thought. But you even have Gildas writing in the sixth century during the Anglo-Saxon invasion/migration to Britain. By the end of the early Middle Ages, you have distinct regional monastic styles emerging.

To sum, in terms of new texts produced that exist to the present day, I’m not sure there are that many fewer texts from the period immediately after the fall of the Western Roman Empire compared to after. And if the drop off is severe, I would not call the period “dark” even in terms of a written record.