I am afraid this answer may not be as comprehensive as you would like, but it ought to answer the core of your question.

Put in the simplest of ways: they didn't leave any written texts themselves, and only ever seemed to interest their contemporaries during hose times that they were involved in the politics of power of their region. In other words, we do know a great deal about them for the period of their arrival in Europe, their first embassies to Constantinople, and their initial wars against Eastern Romans and Merovingian Franks. We know a great deal less about their eventual subjugation and partial christianisation during the Carolingian period, which eventually saw them virtually "passing out of history" and into legend. The word "obor" (a later, probably Slavic development of their name) came to mean a strange, bizarre creature of a dark, legendary past, very much like the name Hun became a giant in German or the name Pict a dwarf-like figure in Scots.

The Avar Empire is notable for its seemingly immense power at the time of its establishment, when it was able to defeat (and, given onomastic clues, absorb) the Gepidae, marshall impressive number of Slavic tribesmen into battle, and challenge the power of the Roman Emperors in concert and on seemingly equal footing with the Sasanid Empire, while being able to launch raids as far afield as Merovingian Germany. Notable also for the almost complete lack of further expansion or continued raiding after the first period of consolidation. We do not hear much about Avar activity between the early 7th century and the late 8th. Reasons for this must remain speculative. Our sources for the time are rather scarce, but one cannot completely deny the impression that the Avar elite seems to have been comfortable (or forced by circumstance) to abandon its earlier Nomadic raid-based economy that is so familiar from other Steppe peoples moving west (Huns, Magyars, Pečenegs, etc.). This may be due to a change in their economy and is as such not unheard of (cf. the relatively consolidated Khazar Empire, also founded by an originally "expansive" nomadic elite). We basically do not know.

For all intents and purposes, the Avars only really take center stage again around 796, when Charlemagne defeated and plundered their empire and carted off the treasures they had amassed. He and his successor attempted to have the remnants of the Avar elite (and the Slavic people under their rule) christianised, but with limited success. Had they been successful, there might well have developed an Avar literary tradition and something like a "medieval" Avar identity (much like it did for medieval Scandinavia after the consolidation of a Christian royal power). As it was, however, the remnants of Avar power in Pannonia were quickly overlaid by rival Slavic kingdoms (the most influential of which was Moravia, wherever one wants to locate it). By the end of the century, the Magyars had established themselves as the new great power in the region. Any budding tendencies to create a lettered, Avar-speaking tradition were entirely eclipsed by these developments, if there had been any in the first place, which is doubtful.

The Avar language itself is only attested in a small number of words, eg titles of nobility. We can assume a large part of the "Avar" populace to originally have been Slavs or Gepidae or a range of other ethne, so similar to the Turkic Bulgar language and tradition, which seems to have been absorbed into a Slavic substrate early on, we may assume that the originally steppe-nomadic elites of the Avar Empire were soon much less "Avar" than their forebears at the time their empire established itself. By the time of its fall, there was simply no one willing or able to carry on the Avar torch, if you will.

Edited for horrbile formatting and typos (my phone and I had a few disagreements.