Hello,
I received my SIN after visiting the Service Canada office. I had to call customer care every day and they gave the same answer that I have to wait unless someone picks up my case. I told them that I have an offer letter but unable to join as I don't have SIN. One of the reps asked me to contact Service Canada. During that time, we had to request online and I booked an appointment online using the below link.
https://www.canada.ca/en/employment-social-development/services/sin/apply.html
I selected to apply in person and the next day I got a call from service Canada and they asked for the details. I explained about my case in detail and said that it is an extreme emergency that I need my SIN to join a job and my savings were getting over. They scheduled me an appointment and I visited Service Canada and got my SIN.
I would say to do the same and explain that you are desperately in need of SIN. Hope they schedule an appointment and you receive your SIN soon. Good luck