You should see a Vascular Surgeon, especially since you have arterial (lost pulse, cold fingers) and venous (swelling) symptoms. It sounds like you have nerve pain, too, but blood vessels are so delicate I think in a case like yours it would be best to consult with a Vascular Surgeon over a Neurosurgeon.
You are geographically fortunate in being so close to Johns Hopkins in Baltimore. As JoMar said, Dr. Freischlag is an expert in TOS, some might say THE expert. I don't know what your insurance requirements are for referrals, but I'd call her office yourself ASAP and ask her assistant what it would take to get in to see her. Make sure you mention you have vascular symptoms and that you have an MRI showing a cervical rib.
There also appear to be some surgeons at UofVirginia who treat TOS:
http://uvahealth.com/services/vascul...utlet-syndrome .
Don't give up until you see the right doctor, especially because you have vascular complications. You may even consider going to the ER when vascular symptoms present and get a workup there (mainly a venous ultrasound to make sure there are no clots) and see if the surgeon on-call will at least speak with you.
I'm sorry you are going through all of this. The doctor shuffle is bad enough, but not being able to live your life and play with your kids is devastating, I know.