I am also the mother of an alcoholic. Every time she tried to access the free detox program at the local hospital, it was full. They only have six beds, and there is so much demand for their services. Eventually, though, she got in. They put the alcoholic's name on a waiting list. It helps if the alcoholic keeps phoning - the staff at treatment centers need to know that the alcoholic is really serious about getting well.
I am assuming your son is also attending AA. This has been a lifesaver for my daughter. There is a huge amount of support available in AA groups and it is completely free.
It also helps if you attend Alanon. You will learn positive ways to handle the feelings you are experiencing right now.
Keep posting here, too.
Love and blessings
Robin