O. Henry can be great fun to read, but I have always urged learners to keep a dictionary close by to clarify some of his finer points of usage. It can be difficult.
The girl in this passage was, at first, glad to have two men in her carriage (probably for safety's sake) but suddenly horrified to discover that they were criminals handcuffed to each other. She was bewildered that a man so outwardly appealing as Easton was handcuffed, not because of any action that he had undertaken.
You say "Relaxing is opposite of distress." That's not exactly correct. The antonyms for the noun "distress" are "happiness, safety and prosperity." For the verb "distress", they are "to calm" and "to please."
Remember, the author is referring to the woman's lips. They parted in a vaguely relaxing manner but still expressing a degree of distress.
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