At that time, lots of the civilisations wrote on clay tablets, and as their cities burned their clay tables were baked and hardened and so managed to survive the destruction.

There were several. Clay tablets from Ugarit, a Linear B tablet from Pylos, cuneiform tablets from Hattusa (in the decades before the city was abandoned), and several records from Egypt during the reigns of Merneptah and Ramesses III.

Largely archaeological finds that yield tons of forensic evidence. Plus there are actually plenty of written records that have survived, albeit with major gaps. The Akkadians wrote on stone tablets, thousands of which have survived. Mostly they are banal transaction records (which still tell us a lot!) but there are many others, including some court records from ancient Egypt and elsewhere.

More specifically, the Akkadian language with cuneiform on clay tablets became the lingua franca of the late Bronze Age, long after the decline of the Akkadian empire. The Bronze Age collapse dates to about a thousand years after the end of the Akkadian empire, but it was still the language used for diplomatic purposes as far away as the Aegean Sea and the Egyptian imperial court. A number of royal and scribe family archives have been found, containing thousands of these clay tablet letters.

> but it was still the language used for diplomatic purposes as far away as the Aegean Sea and the Egyptian imperial court.

And also the language in general use in southern Mesopotamia, presumably subject to historical changes.

Note that while the various states in that region didn't call themselves "the Akkadians", they did tend to claim the title by giving their king the very traditional royal honor "King of Sumer and Akkad". By the twelfth century BC, the title was attached to Babylon.

The main literary primary sources are monumental inscriptions in Egypt and a cache of diplomatic letters from some of the cities that were destroyed in the Bronze Age collapse.