Look at the sentence fluidity, sound it out. Which sounds 100 times better? A. That is because there is no need to add most, sincerely already adds a note of "most sorry".
A. <em>Carolyn and Judy were sincerely sorry about the ticket mix-up</em>
Explanation:
This is the correct answer choice since it wouldn't make since to say 'most sincerely' . They already included their adverb, so there is no need for further elaboration.