After a bit of racking my brains again I came up with a few examples from Greek/Roman myth, though not all are writings by Gods.
1. The Golden apple of Eris, thrown onto the banqueting table at the marriage of Peleus and Thetis, it bore the words “for the fairest” and prompted the competition between Aphrodite, Hera and Athena, the judgement of Paris, and the Trojan War. So not only could Eris write, but the Gods could also read.
2. The Siblline prophecies, written by the Sibyl of Cumae and offered to Tarquinius Priscus, King of Rome, who refused to buy them, causing her to burn six of the books, forcing the king to purchase the remaing three at the original price. The Cumaen Sibyl was offered a wish by Apollo in return for sleeping with her, and she asked for as many years of life as grains of sand that she could hold in her hand. She then rebuked him, and so Apollo allowed her to grow old without eternal youth, so she grew withered away to nothing.
3. According to Euripides, Phaedra, the wife of Theseus, left her husband a suicide note accusing Hippolytus (Theseus’ son and her step-son) of trying to seduce her.
4. Proitos, king of Tiryns, sent his father in law, Iobates, King of Lycia, a letter requesting that he kill the letter bearer, Bellerophon. Proitos’ wife had had her amorous intensions rebuffed by Bellerophon, and then told her husband he had tried to rape her, causing Proitos to seek his death. Iobates sent him against the Chimaera, marauding pirates, and the Amazons but with the help of Pegasus Bellerophon defeated them all and Iobates gave him his daughter in marriage.
Not sure how the final one fits in chronoligically with Kadmos.
Strange how all these relate writing to misfortune through failed loves.