Thanks all!
Hotstreak, our mulch was free, from a neighbor who owns a tree service. Pretty sure there was already some kind of mold in it because after spreading it over the patch (~100 wheelbarrows worth), I had a bad cough for two weeks. When we were shoveling it around, little puffs of something that looked like smoke were impossible to avoid.
As far as watering, we haven't had to do much because it's been exceptionally rainy (and humid) for the last 3 weeks or so. Prior to that, we watered with the hose on the soaker setting, at the base of the plants.
Last night I removed the infected leaves and sprayed two plants with Neem oil, as a test. I'll check them tonight and if they look fine, I'll spray the rest of the plants, and we'll see where that gets us.