Look at Greek during the Byzantine Empire. It evolved slowly, very slowly, especially in written form, although pronunciation changed faster. Even as late as the 20th century katherevousa was a viable challenger for the title of 'official' Greek.
I would guess that Latin, in a continually unified Empire, would be the same - an Imperial core language spelled largely as the classical language, but with major phonetic shifts. Yes, auxiliary verbs WILL come in (when 'vocabo' I will call and 'vocavo' I have called are indistinguishable in speech, you need alternate forms of either the future or past or both; and the erosion of gramatical endings means far more prepositions will be used). However, the massive simplifications that happened in Germanic conquered areas, in particular France and Iberia, wont have happened.
Also, the Roman practice of moving people, especially soldiers, around, means that a common core of Vulgar Latin will be used by everyone in the West, even if their speech at home/in village markets is further from Standard.
Given the extensive corpus of Latin that the Vulgate Bible represents, I think the official language isnt going to stray very much farther than that.