Look at the ERE. Byzantine greek evolved slowly, from Classical Greek, but it was still recognizably the same language. The pronunciation changed a lot, but the spelling and grammar didn't nearly as much
The difference is that Greek doesn't have the same tendency toward dropping final vowels that Latin has (and Latin doesn't have as strong a tendency as Germanic languages). In the Romance languages, the blurring of final vowels and the loss of final consonants came about as a result of the Latin stress accent; Ancient Greek did not have stress accent but pitch accent, and in the other direction Germanic languages had a stronger stress accent, earlier in the word.
Tellingly, there was convergent grammatical evolution in the Western Romance languages: two genders, no case distinctions except in pronouns, a move from Latin's subject-object-verb word order to SVO, the development of definite and indefinite articles. These happened at a time of limited communication across the Romance-speaking world - though not zero, as seen by the fact that where communication actually was zero, as between Romania and the Western Romance world, the differences are more extensive.
So would TTL's *French be more akin to OTL's Occitan?
I doubt it. Occitan still shares some features it got from French. Although the biggest changes from Latin to the Romance languages would have happened regardless, many innovations began in northern France and spread from there in waves, influencing Occitan, Catalan, Spanish, and Gallo-Italian. The development of the front rounded vowels /ø y/ is a French innovation that spread to Occitan and Gallo-Italian (and probably also to Dutch, which underwent a French-like /u/ -> /y/ change in addition to Germanic umlaut). The loss of intervocalic /b d g/, which affected Spanish, started from French as well. The lenition /k/ -> /ts/ (eventually /s/) before /e i/, as in the pronunciation of the letter C in English, originated in French, and is atypical; usually, if /k/ weakens before front vowels, it's to /tʃ/ or something similar, as happened in Italian and many non-Romance languages (Sanskrit, English, Swedish, Mandarin).
If you want a model language for how Romance languages would've looked if the Western Roman Empire had survived, look to Italian, which resisted French innovations. Italian also had its own set of innovations, for example more extensive leveling of consonant clusters than the rest, and if Rome had been able to project power continuously, it's likely those would've spread more widely. In OTL, they only really spread north to the La Spezia-Rimini line, with some limited influence on Gallo-Italian (/k/ -> /tʃ/ before front vowels, and not /k/ -> /ts/).
This is especially true if Latin had truly died in Roman court use (and morphed into Italian), rather than being preserved in church use as in OTL. Spanish and French leveled a lot of consonants, but then restored them under Latin influences - for example, Old Spanish was happy to say perfeto, but then switched to perfecto under Latin influence.
Of course, the written language would remain more conservative, and I can see people try to pronounce consonants that disappeared in Italian (plano, not piano). Then again, English has undergone some massive changes in pronunciation even while maintaining a continuous centralized government with fossilized spelling. Nobody tries to pronounce the L in talk. Even obviously foreign borrowings get vowel-shifted based on English spelling pronunciations.