What does Jeannette do to show that she is finally losing faith in Rex?

She does a lot of things. Saving money to move to New York is one of them.

On page 188 of the novel, it is clearly evident that Jeannette is putting effort to express the idea that she is finally losing faith in her father. One way she does so is by asking her mother to leave him. This is has turned to be a shock father for Jeannette’s mother because Jeannette was always her Dad’s last defender, who pretended to believe all his excuses and tales, and to have faith in his plans for the future. This is one of the many ways she tries. Later in the novel it is also evident that she has plans to leave the family behind and move to New York City which again has the strong signs that she is losing faith in her father again.