Info you give is good, though a little limited.
My suspicion is that it is the disc causing your problems.
Traction might provide temporary relief.
Good muscle tone in the neck and back are essential.
I am not precisely sure where the origin of the problem is but if you have a disc impinging on the nerve roots you will have such problems.
Typically, however, it is C7-8 that give sensory ennervation to the medial hand where you are having problems.
The long thoracic nerve roots come from C5,6,7.
However, I would question a brachial plexus injury as the nerve fibers from the long thoracic and ulnar nerves are pretty far apart.
I STRONGLY recommend you see a GOOD neurosurgeon ASAP.
When there is lack of function, weakness, it needs to be dealt with. I agree with the steroids, though epidural inection may be more efficacious. Ice. Physical therapy may provide some benefit but in your situation might also cause further damage so if it makes your symptoms worse you should think about that.
I never refer to quackopractors
--J/K
I have seen them do more damage in such situations but YMMV.
IMO if you are not getting relief soon or are having worsening symptoms you are probably going to need some sort of surgical procedure.
See a neurosurgeon but find a good one. Doesn't hurt to see what they say.
ETA: Oh yeah, you need an MRI of your neck if not already done.
Bud, this shit needs to be dealt with, you have already waited too long.