The book mentions Sai Baba would literally write his own story. How did he do this if he wasn't a writer living with the author?

This appears to be a misinterpretation of a metaphorical statement. Chapter 2 clarifies that Sai Baba was not suggesting he would physically write the book. He told the author, "He is merely a pretext; I myself shall write my own story!" and later, "Then I myself shall enter with 'I-ness' and write with my own hand." These statements signify that the author, Hemadpant, was to act as a humble instrument. By dissolving his ego and surrendering to Sai, the inspiration, content, and true authorship would flow directly from Sai Baba himself. As Chapter 2 explains, the work is accomplished by the one to whom it belongs, with the human writer only serving as a pretext.