I finished the story mission Double or Nothing in Need For Speed Payback while disconnected from Xbox Live. I confirmed on TrueAchievments that it is earned from finishing that mission with no other requirements. Would replaying this mission allow me to receive the achievement. Likewise, is there an achievement for completing Chapter 4: Open Skies? If there is, would it be possible to re-earn that too?
In contrast to what TrueAchievements states, it is necessary to finish the Double or Nothing race and complete the next mission where you Meet the Broker to earn the Double or Nothing achievement. This was not an issue with Xbox Live, but rather that I didn't complete enough missions to close out the chapter and earn the achievement. This also answers my question of whether there is an achievement for finishing Chapter 4: Open Skies. The Double or Nothing achievement meets both criteria.