A-ha! I plugged in the Adafruit shield, and measured voltages. All wires measured between 3 and 3.15 volts. HOWEVER! The TB6600 measured ~5V and ~7V between + and - on the pairs. Clearly, the motors don't like that at all.
I played with the DIP switches on the TB6600. They were originally configured for 8 micro steps / 1600 pulses per rev and 1.5 amps. I changed it to 0.5 amps and 1 micro step / 200 pulses per rev. The noise is quieter! But still very much there. The Adafruit shield is completely silent. I don't understand the reason for the voltage differential. Is there any way to change it?