As you say, it is definitely possible to retire to France. Though if you do that, you then have to provide your own health insurance (usually through some sort of expat policy - and those can be expensive).

You may want to contact AARO (Association of Americans Resident Overseas) AARO - Association of Americans Resident Overseas - Home and/or the ACA (American Citizens Abroad) ACA American Citizens Abroad AARO offers expat insurance to its members, and both associations have information about tax and other matters of interest to US citizens living abroad. There are lots of retired Americans in AARO and maybe someone there can help. (Just be aware that most associations in France are on summer holidays - you may not be able to contact them until September.)
Cheers,
Bev