Geez, we all love you, already! ;)
Many others know the Loire better than I do, but I'll start:
Chinon is good for a couple of hours at the chateau site, now mostly in ruins, with a killer view over the town and valley. Haven't been to Villandry in many years but I think the gardens are the main attraction there. On that side of Tours, Azay-le-Rideau is in a beautiful setting but you can't see everything. Seems like your first day makes sense given your timeframe and lodging.
Then, which others to see? Well, duh, Chenonceau, when it opens to minimize crowds. Then I'd go up to Amboise for the chateau and also the Clos Luce, where Leonardo spent his last years. Next day, I'd say further east to Blois, whose chateau has four distinct architectural styles with associated history, then Chambord in the forest, the biggest of them. If you have time, Cheverny is nearby and beautiful. If you're interested in Darwinian struggle and aren't too sensitive to smells, catch the 4:00 feeding of the hunting dogs at Cheverny. Or head back to Chenonceaux for your own tastier dinner.
I'd visit two chateaux a day, three at most. All the ones I listed east of Tours are pretty close together. The countryside is wonderful. You're experienced travelers who know how to have a good time, so have one!