First of all, keep in mind that I am unable to diagnose you because I am unable to examine you, this forum is for educational purposes.
The symptoms that you describe could be the result of Carpal tunnel syndrome (CTS), but I would suggest some more testing to rule out a structural lesion in the spinal cord. CTS classically results in burning pain in the wrists at night and weakness and pain with certain tasks (often typing, holding the steering wheel, holding the phone). However, CTS can also result in a wider distribution of arm pain, like you describe, that comes on at night. Weakness in the hands and atrophy of the thenar emminence (muscles at the base of your palm) can also occur in severe CTS. It is odd that you would have both hands affected at once if it is due to CTS. Another possible explanation is a structural lesion in the cervical spinal cord (demyelinating, tumor or vascular malformation). These tend to evolve slowly and are worse at night as you describe. I would recommend an MRI of the cervical spine with GAD contrast and to follow-up with your neurologist to see if your symptoms progress.
I hope this has been helpful.
Sleeping on my back does not help. I have tried all normal and several abnormal sleeping positions, and I get the tingly arms with all of them to some degree. Last night I tried lying on my back and positioning my arms at an angle, with elbows straight but blankets and pillows angled under my arms so that they pointed up at maybe a 30 degree angle. I awoke with my left arm extremely tingly, shaky, and weak. I will not try that again.
As you suggested, many of the sites about TOS talked about stretching exercises. However, none of them actually gave any exercises to try -- reportedly because doing the exercise wrong could worsen the condition. Some sites then posted an office where this could be treated -- usually some office in California (across the country for me). That sounds like my chiropractor who kept telling me to come back every couple of days, but I wasn't getting any relief of my symptoms. It's hard to tell who could really help and who is just trying to make a buck.
Do you work on the computer alot??/ check out a book by d"Arezzo about posture and there are evxercises in there too. It is a great little book. TOS is a wastebasket dx only to MDs who dont know about it. It is a real syndrome. The Dr. I saw is Richard Sanders in Denver. I recommend him highly.