To answer your question simply, no, you most likely can't work.
I've never heard of a "foreign executive" visa for France. There is a "scientific and researchers" visa (actually I believe that's the title of the carte de séjour) where a dependent can work. But a normal work visa, even for someone at the executive level yields a non-work visa for the trailing spouse.
The information on the various consulate websites varies (in both content and quality, I must say) but on the US site I checked, it's never really stated plainly about the work status of dependents on "accompanying family" visas. The "scientific researchers and university professor" is the only one I'm aware of where the spouse can work. But for "foreign executives" the correct visa type is a "work" visa, plain and simple. Which consulate did you apply through?
If your French is up to it, you might want to check on the auto-entrepreneur site Portail officiel des auto-entrepreneurs
They have confirmed that you can set up an AE business on a "visitor" visa, and I suspect it might depend a bit on your local prefecture whether or not they'd clear you for AE status, which would give you leave to freelance in teaching English.
The other option is to telecommute back to the US. It's not exactly legal, but generations of trailing spouses have gotten away with it over the years.