The new board hasn't come yet, but a friend gave me a great idea, to power the mic from a GPIO, which powers it off completely when the ESP is off.

Hopefully the new boards will be here soon, but another issue is that I don't really have anything that can measure microamp consumption, so any testing takes days of waiting for the battery to run down :(

I do think these clones are the issue, though. They had a LED I couldn't turn off, so they'd literally shine forever. They don't seem engineered for low quiescent current, so fingers crossed with the new ones.