I agree with Spilner. What afflicts the three major characters--the sheriff, the deputy, and Mildred--is anger at the injustice of the cosmos.
The sheriff is a beloved man in the town, with a young wife and young children, and much to live for. His life is being cut short way too soon.
The deputy is a victim of birth--low intelligence, low socio-economic status, few choices in life.
Mildred, of course, has lost a child in a barbaric way.
Each character responds differently to this injustice.
The sheriff refuses to take out his anger on others: he creates one last beautiful day for his family, a memory that he hopes may sustain them. Not only does he refuse to hurt others, he works to heal those around him with the letters he writes. He offers a model in the movie of how to deal with the cosmic absurdity.
Mildred feels terrible guilt for saying something that all of us have said in some form or other, but in her case, her curse came true. She lashes out at a symbol of authority and control (the police; the sheriff) because she has decided not to hurt herself and she cannot reach God or "the universe" or whatever is operating at the grand scale that controls our lives. In the process, she dehumanizes those around her, refusing to acknowledge that her son is also suffering, refusing to see the sheriff as a human being who is also hurting.
The deputy has gotten his revenge on life by using his position of power to inflict the injustice he feels on others.
The sheriff's letter to him changes him. His change helps Mildred reconsider her blind rage.
Chances are excellent that the deputy and Mildred will not harm the fellow they are headed out to find. Their grief and rage have been transformed, and their emptiness has been somewhat filled.