Just wanted to let you know I'm feeling similar to the way you are feeling right now. I just graduated with my BSN and I'm trying to find a job in the Boston area. I also have Social Anxiety Disorder. I can definitely relate to you when it comes to the financial piece of trying to afford mental health treatment. I currently cannot afford my medications, but it is a little motivating for me to push myself harder to get a job by putting myself out there, so I can afford those medications. It's been a constant struggle for me all through nursing school to just get myself to walk into a patients room and say hello and take their vitals. I found that it was very, very hard at first because I expected every patient to be mean and horrible to me (I'm a child abuse survivor and have some PTSD as a result). But I found that each experience I had gave me a little more confidence and now I don't have much anxiety at all before going into a patient's room. In my four years I only had one patient who was kind of a jerk to me, but thankfully I had an amazing instructor who helped me deal with the patient and the aftermath of dealing with him.
Through my therapy the last few years I've learned that a lot of social anxiety can be improved through exposure therapy. It takes time, but you can slowly give yourself little challenges to build your confidence and move on to more anxiety provoking tasks. As far as looking for jobs goes, even applying online was anxiety provoking for me, so what I did was I had my sister sit with me while I worked on an application and if I felt nervous about something I would tell her what I was feeling. Having her reassure me really helped me relax and push forward with the application.
As far as references go, I think you would be surprised how many of your instuctors and professors remember you. I remember I was SO anxious about asking for references but once I sat down and emailed each person individually (with my sister next to me for support) I managed to send out the emails and got great responses back (Of course I remember you, etc). It's okay to acknowledge to the person that you were very quiet during school, chances are they will understand that (if they are an empathetic nurse as they should be, they will understand, if not, they probably aren't someone you'd want a reference from anyways).
Just remember the challenges you have faced regarding your social anxiety. That can be extremely empowering as it makes you more empathetic and compassionate, so don't let anyone ever tell you otherwise. You may have to work harder than others when it comes to coming out of your shell, but you have the potential to be a great nurse, just like anyone else.
I'm always here to chat, you are not alone!