Out of the Dust won the Newberry Medal largely because of its unique free verse format and strong emotional themes. The lines
"I am awkward with him,
and want to be alone
but I am terrified of being alone"
are strong examples of both. In reviewing the lines above, it is important to remember that Billie Jo has lost her mother, her baby brother, and the use of her hands through the accident with the pail of burning kerosene. On the surface, the loss of all of these was her fault. She was the one who threw it on her mother. While it saved the house and she was innocent in her motives, the guilt is unbearable. Her normal escape of playing the piano has been taken from her, so she is left alone with her father in a house that seems filled with only bad memories.
Billie Jo has lost her father. He has been consumed with grief and despair. Due to this, he has pulled away from her. Billie Jo is struggling with the desire to hate her father for leaving the kerosene in the kitchen. She keeps wondering if she will ever forgive him. Add to this emotional landslide the fact that they are both locked inside the Great Depression, in the middle of the Dust Bowl, Billie Jo feels that they are being eaten away like the dust. They are consumed by their own guilt and anger at themselves, each other, and the situation.
The lines referenced in this question speak to the emotional crisis that Billie Jo is experiencing. She has not made the decision to run away, but she is feeling restless. These lines express her desire to leave the situation, which she might do but for the recognition that he is the only family that she has left. This is what ties her to him and makes the situation all the more unbearable.
In the same entry, the first stanza talks about the changes Billie Jo sees in her father. He looks like her father. His actions are that of her father, but he is no longer her father. The lines at the end of the entry speak about the void and the pain. The loss of her mother seems to be everywhere and no matter how they move or where they sit, she is gone and they can feel it. She talks about her hands and how she hides them. Billie Jo does not indicate whether she hides them because she feels he is blaming her for her mother's death, or because he is blaming himself for the burns on her hands. She only knows that he is staring, and it makes her uncomfortable. This will lead her to leave.