There are plenty of sources throughout the late antique/early medieval period. Even ignoring letters, poems and other literary evidence, just for the sixth century, I can think of:

• John of Biclaro and Isidore of Seville's Chronicles/History in Spain from the late sixth century
• Gregory of Tour's History of the Franks from the same time in southern Francia
• Victor of Tunnuna's Chronicle from Africa a bit earlier
• Liber Pontificalis from Rome, which began to record contemporary lives of popes in the sixth century. Very focused on the papacy of course, but lots of incidental details are mentioned and we have actual documents from Ostrogothic/Roman governments to supplement our knowledge of Italy in this period.
• Theodore Lector, Zosimus, Marcellinus Comes, John Malalas, Procopius, Agathias, Menander Protector, John of Ephesus, Evagrius and more from the Eastern Roman Empire, covering everything from Africa/Spain to Mesopotamia.

Some of them are obviously more reliable than others, but even if they are not accurate, they are still useful sources as they are examples of how certain people thought in a certain period, which can be very rewarding when looking at cultural history. These historical traditions continued everywhere but the Eastern Roman Empire in the seventh century, but even so there are plenty of surviving letters, hagiographies and later histories to fill in the gaps. Admittedly many historians don't tend to treat religious texts as sources for 'secular' history, but the trend right now is to move away from that. There might be less contemporary historians writing in the mould of Tacitus/Ammianus Marcellinus, but we have more people writing about local saints and many, many surviving charters, both illuminating a different part of society compared to traditional secular narratives, so it's a case of losing some details but gaining more knowledge in other areas. In any case, by the Later Roman period secular histories were quite rare, so it wasn't a dramatic change when people started to write mainly ecclesiastical histories/chronicles in the post-Roman world.

Well it depends on where you are interested in but in a lot of cases the fall of Rome did not significantly alter how things worked (at least in France), instead of the Romans being in charge it was just someone else. But for France you have Gregory of Tours "History of the Franks", then around when I study which is the Carolingian period you have lots of different sources from letters, capitularies, poems, etc. The sources never really "become quieted", at least if I understand what you are asking. If you want to actually check out the sources there is a book called "A Source Book for Medieval History" that is translated by Oliver Thatcher and you should be able to find it if you have access to a college library. If you can read latin there is always the Monumenta Germaniae Historica, which contains almost anything you could ever want (but its frustrating to use and its all in latin haha)