Sorry to hear about your mishap- though imo it’s part of the fun of a touring holiday! I’ve just got back from a 2300 mile tour of Europe.
I would say if you’ve got breakdown cover, use it. If not, I would get the car properly inspected by a local mechanic- you can then decide on what to do depending on what the problem is, how far you have to go etc.
What I wouldn’t do is to risk catastrophic failure of the engine by continuing to drive long distance. There’s obviously a problem there, and it’s not like you’re nursing the car a few miles down the road to the local garage. You’ll get some technical advice from knowledgeable forum members, but they’ll only be guessing at what the problem might be.
I arranged breakdown cover with the AA for £60 for a week. One of the questions they asked is whether I was already abroad ( when I called to arrange cover) so presumably it is possible to buy cover when you’re away? That may a an option worth looking at.
Where are you in France?