Words of encouragement: I was able to book my biometrics in Coimbra with patient diligence. As noted, cancellations do open up from time to time, particularly at certain hours of the day and week. Next-day Azores appointments often open up in the hour before midnight, Portugal time. But note that you can not cancel an appointment on the same day, so if you book it, you’re stuck with it! And the immigration people will be cross with you if you miss an appointment without a good excuse.
One of the big law firms appears to manually reshuffle the appointment calendar during the mid-morning of Portuguese business days. You can often find plum spots if you check at just the right moment. I found a Coimbra appointment two weeks ahead of time–plenty of time to book travel and all that.
Good luck everyone!