how to find lodging for a one-month stay?

I haven't done what you are proposing, but I suggest using Homeaway.com and Airbnb. If you choose dates over three weeks long, Airbnb will provide monthly rates.

Have you looked at gites? They are specifically for longer stays (at least a week) and are available all over France.
https://www.gites-de-france.com/en/

Don't neglect the critical step of checking your reach by rail for day-trips once you identify a possible base. The rail lines in France tend to radiate from Paris, so it is often necessary to travel back to Paris and out again to get between two places that are not all that far apart but are on different rail lines. To make it more annoying, you sometimes have to change train stations in Paris, which really adds to the travel time.

Keep in mind that trips involving TGV (express) trains may be quite expensive if you don't buy the tickets well in advance. Fares on TERs (slower regional trains) do not increase as the travel date approaches, but you cannot cover as many miles in an hour or two if you're on a TER.

You can check schedules, the need for transfers, and fares on the SNCF website. I prefer to use the Deutsche Bahn website if I'm just taking a preliminary look at schedules (not fares).

Booking.com also offers apartments, so check them as well.

Airbnb and booking.com. Booking has some that airbnb doesn't have and also there is a lot of cross listing. If you find a place on airbnb, check that it's not cheaper on booking.com (and vice versa).

Read the reviews carefully and only pick a place with a lot of reviews so that the subject of the internet will have come up if it's no good. In smaller cities in France especially, I've found that some newer or low traffic airbnb hosts use a very slow service that is unusable for working at home. This isn't normally the case in busy airbnbs since if it was an issue at one point, the hosts would see the feedback and switch to keep business coming.

One trick with wifi is that if you are in a place that turns out to have terrible internet, look for other networks that are sfr fon. You will be able to connect to these if you are a sfr client or if you buy a sfr access pass (30 euros/30 days). If you're staying in an apartment building, one of your neighbors will have sfr and have it shareable as the box default is for sharing it to be on.

Edit: Try to stay in a city served by Ouigo so that you have the option of discounted tgv. The strategy would be to stay on one of the longer routes ie somewhere between Paris and Nice, Paris and Marseille, Toulouse/Bordeaux and Paris, etc.

I have found my apartments in France on both VRBO/Homeaway and booking.com. No problems with any of them. My VRBO apartment in Paris cost exactly what was stated on the website.

I've done both hotels and airbnb. I like hotels because I don't have to worry about housekeeping and I'll probably have air conditioning. Sometimes hotels will give you a special discount for staying a month,

But if you absolutely need a kitchen, few hotels offer that. I personally don't like cooking on vacation.

For a longer stay, I like Gites. Make sure you know whether you need to bring your own bedding or not? It's not uncommon for longer stays in Europe. If so, I'd just stop at a store there rather than pack them. Afterward, I'm sure they'll accept your donation of clean linens.

Hi Bill, My husband and I are doing the same! I have begun research on towns and thought I'd share!

Here is a link: https://community.ricksteves.com/travel-forum/france/month-long-home-base

Hi Bill, Just circling back to see which town you decided on? I am still on my search for a town to have a similar work remotely for a month experience.

We use Homelink.org.

We have done this many times by exchanging our home! 14 exchanges in about 13 years. It's nice because you are in a home, it doesn't cost you anything, and your home is being cared for and lived in while you are away.

Take a look at house/pet sitting. We like www.trustedhousesitters.com. Also often overlooked is www.sabbaticalhomes.com. We rented an apartment in Rome for 18 months with the latter and frequently have folks from Trusted Housesitters take care of our home when we travel for a long period.

Google digital nomads and slow travel and you'll find communities of people doing what you're contemplating. We've been staying about a month at a time in one location for the last several years. We do this after other travel--- sometimes a land tour, a safari, a river cruise, etc. and want to land somewhere and just hang out and soak in the atomsphere and live like a local to the extent possible. Have fun!