Pilgrim House does NOT have showers available. The place up the street that used to have showers for rent, is now Casa Ivar - Ivar's office. There are no showers there either.
IMHO and experience, getting a room, even a bed at an albergue is the...
I found two entries on Gronze forum (left by two different Spanish pilgrims) saying that they were able to take a shower at Albergue Azabache for a small fee. This would be very close to the Cathedral.
https://maps.app.goo.gl/WmhSVfnFBbXdfwLt5
There is a public bathroom as you face the cathedral behind you to the right down the steps and against the wall there. There are two or three restaurants with tables along that area so look along the wall for the doors.
This site is run by Ivar at in Santiago de Compostela.
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