Hi, sulu, welcome back! You lasted longer than many do with your "no more Caminos" pledge.
No shame in succumbing once again.
I would join your recommendation for staying in one of the two albergues outside town and a few kms up the road. I think most of us who managed to do that stayed in Villa Castora, but that's the one on the left, so Alan you either have the name wrong or were looking at it from the other direction. Both are run by the Comunidad de Madrid (the Greater Madrid government). And both are on a road with the name of "Dehesas" so that may add to the confusion. Sulu's recent experience confirms that the one on the right is also open to pilgrims.
Here's the website for the one on the left, the Villa Castora
http://www.madrid.org/cs/Satellite?..._FA/JUVE_alojamiento&rootpageid=1152517028736
And the one on the right, a bit further up, Las Dehesas
http://www.madrid.org/cs/Satellite?cid=1142322453978&language=es&menuIzquierdo=true&pageid=1142322433572&pagename=PortalJoven/JUVE_Alojamiento_FA/JUVE_alojamiento&rootpageid=1152517028736.
PRices are the same at both
http://www.madrid.org/cs/Satellite?cid=1142322437488&language=es&menuIzquierdo=true&pagename=PortalJoven/Page/JUVE_contenidoFinalMenuIzquierdo&rootpageid=1152517028736
With your credential, you don't need a youth hostel card. When I was in the Villa Castora, there was a huge group of school kids, but we arrived at their afternoon opening time (5?) and were given private rooms off to the left, far from the noise and the kids. I was told they keep those rooms for pilgrims, but I don't know that for sure.
Buen camino, Sharon! Laurie