I experienced the same symptoms - cold and damp in the morning - distortion. Warm and dry (sometimes later in the day) and all okay...
A thread on miata.net relayed that the centre cone 'solenoid' in the speaker was found to be rubbing on the outer core (I have not described this very well) and this was sensitive to temperature, ie when it was cold the clearance was tight (due to an imperfection) but was okay when the heat of the day expanded the clearances. I was about to explore this issue when all of a sudden the summer arrived and I have not had a problem so far. I still think moisture on the speaker, etc may play a part but this is just a hypothesis that I have not tested yet.
The fact that both speakers on your car suddenly became affected after washing, when of course water enters the door skin, lends support to this though. There is nothing you can do to prevent water ingress as the window seals are not waterproof. There is a water shield on top of the main speakers to prevent direct contact...
Your thought about the electrics/amp may have some validity - good thought. My problem also disappeared, coincidently, with me solving a minor leak into the cockpit from the screw that holds the external windscreen valance on the car following a screen replacement. see last couple of pages on this thread where I described my resolution:
http://www.mx-5.com/Forum/Room/Topic/?topicId=3457&replyId=129684#r129684
Keep us informed - as I said- I think I will be revisiting this...
David