I also have throbbing and aching hand pain. It is rather severe...
I would not characterize it as "burning" and I do not have numbness or tingling. What I have gathered from numerous docs, is that it is NOT very characteristic of TOS. I have started looking into other explanations/causes for the hand pain...

I would be very interested to hear what Dr. Donahue says about it.

Otherwise, I took gabapentin for about a year, then I tried Lyrica for also about a year. I didn't think they helped. More recently, I have tried amitriptyline. Amitriptyline is a tricyclic antidepressent that is often prescribed off-label for neuropathic pain. I don't know if it helps the pain -- but it does help me SLEEP. It allows me to sleep through considerably pain.
You might want to consider it as an option to get through the nights.
I get the sleep effect with a relatively low dose.