The word "obstinance" appears to be used incorrectly in most of the sentences you've provided. However, the most correct usage of the intended word "obstinacy" (which means stubbornness) is in the sentence:
"Sheila's obstinance meant she refused to do her chores even after Dad told her she'd be grounded if she didn't."
While "obstinance" as a word doesn't appear in standard English references, the context suggests it relates to obstinacy. If "obstinance" was intended to mean "obstinacy," then this sentence conveys the correct meaning effectively.
The usage in the other sentences is incorrect either due to the context or because they use "obstinance" incorrectly regarding its meaning.