In Karen Hesse's Out of the Dust, written like a series of diary entries, Billie Jo Kelby recounts in her entry for April 1935, titled "Let Down," how she had been invited to play the piano for graduation. But due to the scars from her hands having been burned, she was unable to get her hands to work. All she could do was sit at the piano, staring at the keys until "folks started to whisper" and "even Miss Freeland started to cry." At this point, Billie Jo also reflects that if her father would ever go to Doc Rice about his skin, she might be able to get advice about healing her hands. But since she knows her father won't go to Doc Rice, she further reflects, "I think we're both turning to dust."
Dust is often used as a metaphor for death, such as in the metaphor the Anglican church uses during a funeral service when delivering the deceased person's body to God: "ashes to ashes, dust to dust" (Book of Common Prayer, "The Committal"). Hence, in saying "we're both turning into dust," she is reflecting on their ruined, unhealthful state in need of a doctor and thinking about just how much the dust is destroying their lives. While Billie Jo and her father are further from being literally killed by the dust storms than others, she can't help but notice how much the dust is ruining their physical beings, which makes her draw a comparison between their ruined health and being turned into dust, just as the dead are turned back into dust.
She particularly associates herself and her father with death due to the accident that happened July 1934. Her father had left a pail of kerosene by the stove, which her mother mistook for water, causing a major fire. It's due to the fire that her mother soon died of major burns and Billie Jo's hands became burned.