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Praise be to Allah.
If the situation during your Hajj was as you describe, then you do not have to offer a sacrifice for not staying overnight in Muzdalifah because you are excused, and your Hajj is valid.
Shaykh ‘Abd al-‘Azeez ibn ‘Abd-Allaah ibn Baaz, Shaykh ‘Abd al-Razzaaq ‘Afeefi, Shaykh ‘Abd-Allaah ibn Ghadyaan, Shaykh ‘Abd-Allaah ibn Qa’ood.