I'm curious as to how the transition from Latin to Greek for official purposes took place. Was it a slow evolution or could individuals perceive a sudden shift in the Eastern Roman Empire?
You wouldn't have a problem because Latin was still used, particularly during Constantine and his sons' reigns. Latin was used quite prolifically for administrative purposes at the time. However as the administrative demarcation between east and west developed through the fourth and fifth centuries there was a shift to Greek. This didn't make the Empire any less Roman it was just easier to use Greek as more of the Empire's subject spoke Greek. The Greek spoke by people in Constantinople was Koine (almost like Common Greek.)
Latin was still an Official Language of the Empire during the reign of Justinian (527-565).