Having met and worked with a number of people suffering from fibromyalgia, the thing I've noticed they have in common is excessive muscular tension, and I mean off the scale, accompanied by high anxiety. Nobody knows what causes the illness, or how to cure it, but it seems to me that dealing with these two areas must surely help.
The difficulty is that by the time a person has got to a point when all movement causes pain, a weekly Alexander lesson cannot compete with the thousands of inefficient actions and emotions in the rest of that person's life. Every single activity needs to be learned from scratch; every troublesome thought process examined - it's a big challenge.
If it were me, and I could afford it somehow, I would try to have a course of daily Alexander lessons. Have a taster first, to make sure, then book into a hotel or B&B near a good teacher and aim to have at least one lesson with them per day.
The hotel environment should have none of the normal distractions and demands of life. Indeed, you could create the perfect 'minimal' routine, making sure your temporary home suits you completely, including bringing your own bedding if needs be. Then spend the week doing absolutely nothing - unless it's designed to help.
I live in a beautiful part of Dartmoor which lends itself perfectly to these sorts of holiday. Working from home, I'm more than happy to see clients twice a day, thus replicating the kind of intensive learning favoured by FM Alexander himself.
I wish I could say that the Alexander technique could cure fibromyalgia - I can't. I've not worked with enough people, or for long enough, and, as far as I know, there have been no clinical trials. All I can say is that when I do work with these people they like it, they respond well and it all seems to make very good sense.