I have never gone on an organized tour, so my views are biased. It's fun to plan your own agenda and route. Have you considered simply picking an area of France that piques your interest and going there? So if you are into wine, choose a wine-growing area, say, Burgundy, Bordeaux, Alsace, Champagne, or the Loire Valley, etc. Then take a few books out of the library and/or search the internet, make a few notes, and voilą! You are on your way to a tour!
You will need maps. The Michelin maps are excellent, but are not as easy to find on this side of the pond. I get mine at a specialty map shop, but even so, they don't always have the ones I need. Know that you can buy Michelin maps easily in France.
You will likely land in Paris, and from there you will probably want to take the train to your starting point. You can bring bicycles on trains in France, but not on every train. Your travel agent should be able to arrange your train ticket before you arrive.
It sounds like your existing bicycle would be OK. Just have a bike store check it over thoroughly and replace any worn parts. This is the perfect opportunity to upgrade components. Some airlines will allow you to take a bike overseas at no charge, but even if you do have to pay, it will be a lot less than buying a bicycle overseas.
You do not have to be an expert repair person to tour solo without a SAG wagon. If your bike is in good condition before you go, and you can manage to fix a flat, that should cover 90% of your maintenance needs. If you can take in the slack on cables, you are 95% covered. There are bicycle stores in many towns if you get into a real bind.
I have never had trouble finding places to stay in France, except during holiday weekends. But if you are going in June, I doubt you will have to worry. In most towns you will find clean, relatively inexpensive two- and three-star hotels.
On my best bicycle trip in France, I arrived in Paris and took the train to Wissenbourg in northern Alsace. For five or six days I cycled through beautiful villages and endless vineyards, to a mountain-top monastery surrounded by a wall built by the Celts, and up and down mountain passes. Then I took a train to Switzerland for a long weekend spent with friends. I took the train to Dijon, the capital of Burgundy, and then for three or four days went from famous vineyard to famous vineyard. I also visited a stone-age archeological site in the forest, and toured a stunning castle with a multi-coloured roof. On my last day, I visited one more winery, bought several bottles, and drove directly to the Dijon train station. I entrained to Paris, and flew home with wonderful memories and delicious souvenirs!
Bon voyage!