If it is TOS, you are wise to try and resolve your issue promptly. Symptoms in your little and ring fingers mean there is something going on with your ulnar nerve. Often too, there can be a feeling of a flush of coldness over the top of the hand. There are many websites with pictures and description of the places where the nerve and subclavian artery get compressed. One place is the thoracic outlet, another is your elbow.

As suggested above, neurological studies will be needed (many folks with TOS are not positive here, but I was).

The important thing is to find a doctor who actually treats this. Many won't touch it. In the US they are generally either vascular or thoracic surgeons. The doctor will order tests and then send you to physical therapy for a good long while to see if that fixes you. Often it does.

The therapy is about opening up and stretching the pec's (pec minor especially), and strengthening of shoulder girdle and upper back.