welcome back to our community and thanks for reaching out to us again. As far as I know, with a Wohnberechtigungsschein you are allowed to rent social housing, but you are not entitled to a placement. You have to be proactive on your own and specifically look for WBS (Wohnberechtigungsschein) apartments. Often the owners of social housing are large housing cooperatives. It is best to inquire with them about vacant apartments and have yourself placed on the respective waiting lists.
Unfortunately apartments with the Wohnberechtigungsschein are limited and rarely rented. The few WBS apartments that do get rented are highly competitive and competition among apartment seekers is high. For this reason, I would strongly recommend to prepare all documents for the apartment application and to take them with you when you first visit the apartment.
I will also link our expert @mbeon-fardeen here, maybe he can have a look at your question, correct me and/or add more information. I hope this was helpful and please don't hesitate to get back to us with any further question.
All the best, Julia.