Welcome to the group, Jocelyn. I have bilateral neuro/arterial/venous TOS so I can speak from experience about symptoms and testing. Both of my arms from about mid-bicep down are reddish and I have distended forearm and hand veins. This is likely because of poor venous return at the compressed subclavian vein. If i am not mistaken, bluish discoloration has more to do with arterial problems (not enough blood getting to the arm and hand). I know the "white sign" is arterial as well. It sounds like the redness + visible veins in the chest, shoulder, and upper arm (I have those, too) are primarily venous symptoms.

It seems like a waste to order another arterial doppler test (the one where they put the probe on your fingertips to measure your pulse in different positions), although you should INSIST on getting a venous doppler test to check your axillary and subclavian veins on both sides. I have had specialized ultrasounds that look at the veins with my arm in different positions to see the effect on my blood flow. They can also check for any evidence of clots in a venous ultrasound, which is the most serious complication of venous TOS. It is really important that you get this checked out.