Do you plan to rent a car either for the entire time or each weekend? You mentioned trains, and that will make quite a difference in the day-trips you'll have available to you.
I'd suggest opening Google Maps and ViaMichelin.com and focusing on each possibility, one at a time. Look on Google Maps for possible day-trip destinations near your proposed base. Check driving time on ViaMichelin. If you plan to use trains instead, check travel time on either the Deutsche Bahn or the SNCF website.
Bordeaux is a large, handsome city with much grand architecture. Business-like (I've heard people call it "snobby" or "stuffy", though people were perfectly nice to me). A lot of wide streets. Obviously wine-oriented. Not generally an intimate-feeling place, and I wouldn't call it "charming" overall, though there's a neighborhood or two with a medieval street layout. You can day-trip to St-Emilion, but I think you'd be a bit too far from the Dordogne. I suspect it may not be quite what you're looking for. I think a week would be long enough there (too long for many people).
Aix-en-Provence is much smaller. I only day-tripped to it. I liked it, but that's not a good basis for evaluating its suitability as a month-long base. I'm not sure it's the most convenient place in Provence for what you plan to do; it's a bit farther east than a lot of the little places people talk about visiting. If you're planning to use the train (or buses) for your day-trips, be aware that even with a more northerly base, there will be some small places in Provence you cannot reach unless you rent a car or take a tour.
I know quite lot of people are lukewarm to Avignon. I confess to wondering how many of them actually spent more than a few hours in the city, because I thoroughly enjoyed it. There's a good-sized historic area. You just need to venture off the direct route between the train station and the Palais des Papes. But I am rather easily pleased if you plunk me down in a town with a pretty historic district and a bunch of streets where I'll be about the only foreign tourist afoot. Avignon is a convenient public-transportation hub. I stayed there for about a week, making side-trips to Arles, Villeneuve-lez-Avignon, l'Isle sur la Sorgue and St-Remy.
Antibes seemed suitably "charming" on my day-trip there during a longer stay in Nice. Trains run up and down the coast, making it easy to visit a lot of places. You could hop on a train to have dinner in a neighboring town. To get up to the hill towns without a car, you'd often (maybe always) have to take first a train, then a bus.
The stretch of the coast and adjoining territory down close to the Spanish border is also interesting. There might be enough going on in Narbonne to work for a one-month stay. I'm not sure about that, or about how far you could reasonably travel on day-trips, because I stayed in Perpignan (scruffier), partly due to its workability as a jumping-off point for a trip on the Yellow Train through the Pyrenees.
Isn't Giverny quite small? It doesn't have its own train station.