Have a look at the ACSI card (
www.campingcard.com) and Camping Cheques (
www.campingcheque.co.uk) websites for sites. Use either, and you'll get sites from between 11 and 15 euros per night, including pitch, car and caravan, two people, awning, electricity, and one dog.
Using Camping Cheques try Lac D'Orient, (nearish to Troyes) is set alongside a lake, with Troyes worth visiting (nice mediaeval centre, a bit like York, with very narrow overhanging streets of wooden framed houses), and there's the lake itself, the Lac du Der Chantecoq, which has watersports, and bird-watching.
Further south try Lac de Panthier at Vandenesse en Auxois, alongside the reservoir - another nice site, in a pretty area, and near enough to visit the Chateauneuf at the top of the hill, the Chateau at Commarin, the town of Beaune, and the vineyards of the Cote d'Or.
South again and try the hinterland to the site of the autoroute south of Lyon, either the Vercors side to the east or the Ardeche to the west (loads and loads of sites, there to choose from)
Then south again to the Camargue/Provence area around St Remy de Provence (again lots of sites to choose from in those areas).
You could make a similar itinerary using the ACSI card.
There are good maps on both websites, each indicates the position of a campsite, and leads you to an information page about the site, giving dates when the ACSI card or Camping Cheques are accepted.
Otherwise, in May, why not do as we do, and go 'with the weather'? We never have a real destination in mind - we just head south until we reach some sunshine and then stop - for as little or as long as the site holds our interest, and then we move on. In May you'll have absolutely no need to book (even in Frejus) and can alter your itinerary as the mood or the weather takes you.