I have an idea to make the Alexa not evesdrop.
Have a separate device to which the Alexa’s microphone inputs (2 wires) are plugged. This separate device is programmed to recognize a user-chosen password. When that is heard the microphone is enabled and you use the Alexa as normal.
After a user-chosen period of time the separate device would shut off the microphone.
My Fire TV Cube has 8 microphones in it.