OP, I would think very carefully about the move if I were you. I moved to France a few months ago.
I lived in the UK for years, most of them in London, which I absolutely loved, but with two very young kids, my partner and I both thought we would get a better quality of life in France. We are both French, which explains why it seemed so attractive, and indeed rational, on paper.
My partner did find a good job very quickly, and we moved to a huge house with a big, beautiful garden with a nice pool, in a small town, schools nearby, very good neighborhood, and only a few minutes away from one major city in the South of the country. What more can one ask for, you might think.
But - and that is a huge but: we really are quite miserable here. All our friends live in London. We do feel very lonely most of the time - and we are. We miss our life in the UK so much.
France is a very peculiar country. Well, all countries are, but the French really do things and think very differently than the Brits. Cliché, but very true. We are still experiencing a huge cultural shock - and adjusting is not easy. And we are French.
Everything - and I mean EVERYTHING seems super complicated and very administrative. People are usually not very open, and I think the recent terrorist attacks have changed things for the worst. The general atmosphere is indeed quite tense.
Your thoughts about finding a job in teaching upon arrival is not very realistic I am afraid. Many posts were correct in emphasizing how difficult it is to 'meet the criteria' to be offered any job in France, let alone one in education. If you are not completely fluent, you can forget it. And if you don't have the relevant qualifications (which in French means only the French ones), it is also near impossible. I have a degree from one of the top French Grandes Ecoles and find it hard.
Language is crucial to master if you want to integrate - again this is true I think of every place, but perhaps even more so when it comes to France, a place where language has always held a specific and distinctively important place in public life.
Put simply, French people usually don't speak English, and while they will love your accent, you will not be able to integrate if you don't speak the language, full stop. This alone can be very isolating - I am thinking about your partner here, learning French from scratch is very ambitious, it can be done but it would require a massive investment in terms of time and energy, and not mastering the language can really be an issue.
The school system is quite complicated too - like everything else. You do have lots of good public schools, but you also have to be careful if you live in a big city. Most of the time, well off people put heir offsprings in private catholic schools, and often for good reasons: security is a major issue in many places now. Fees are usually not an issue however, on the positive side...
I realize I do paint a rather grim picture of the place, one which may well be tainted by current experience which is not a happy one. But, having lived in the UK for many years, i also know that France tends to be much idealized over there - many British people spend their holidays in France, or have their second home in a Dordogne village, love French food and wine, ect... and assume they would love moving there because of this. Well trust me, living there is an entirely different game.
On the plus side, and objectively, the health care system is really excellent compared to the NHS (however much the British people love their NHS). House prices are cheaper. Food is good, wine is good. And yes, it is true that many people move here and love it here, or at least have a good life - I just don't happen to be one of them.
I totally understand your fears, and in my views they are justified: with insight, if I could go back in time, I would not make the choice I made 8 months ago. We had a great life in the UK, great friends, not even family but just great friends, and we miss all this every single day, in our huge house in this lovely French town. Let me add that we did not have great jobs in the UK either - in fact my partner has a much better job here, but he still misses the UK crazily.
Take time to think about it, and don't romanticize the move too much - the thing about the kids growing bilingual for example :).
Wishing you the best whichever path you chose to take.