I've got through Dry Season and I'm not at the start of Mild and, for some reason, puddles keeping spawning, expanding and disappearing again over short periods of time. I'm not entirely sure what triggers their appearance, but they seem to always show up in the exact same spots, which leads me to think they do not actually despawn when they "dry up".
I've attached my save file. The world is in slot 1. Some of the spots that keep producing puddles are those indicated in the screenshot and at least 2 more near the Slanty Shanty with the Cottage exterior.
Don't really know. Just wait...? and the puddles will come and go.