I'm a bit skeptical. Is there an account of someone who came to Sai Baba with doubts but eventually became a believer?

Yes, the story of Boss Dharamsee in Chapter 35 perfectly illustrates this. He arrived in Shirdi with a 'clinical scrutiny' and was critical from the start. When Baba offered him grapes with seeds, a variety he disliked, Dharamsee internally questioned, 'If he is a saint, how does he not know that I do not like these grapes?' However, when Baba insisted he eat them and they miraculously turned seedless in his mouth, Dharamsee was amazed. As Chapter 35 describes, 'All his ego dropped away, and love for the saint was born.' This experience transformed his skepticism into firm devotion.