For entering France, you have Insurance and the existing CT… and the Carte Grise (albeit not in your name) … I think all you now need is the document to prove you have legally bought the car… but I’m sure others will chime in with more help.
@Mark_Rimmer will confirm what needs to be done by way of documents from seller to owner and also regarding the CGrise (what, if anything, the Seller and the Owner should do…)
(re Selling/Buying cerfa 15776*01 is one form, staring at me from my study pegboard. I believe that form is still in use… but @Mark_Rimmer will know…)
You’ve time to get it all straight… before you hit France.
and, if you get all your ducks in a row… once you are finally here in France, changing ownership should be easy-peasy…
best of luck.