I have had a fair amount of child related stress. My eldest son, who also has inherited my neuropathy is an alcoholic with Asperger's syndrome, an autism spectrum disorder. He is ill, accident prone, depressed, and lives and works overseas in an unpredictable environment. We have five children, the two youngest have learning disabilities, the youngest has physical disabilities as well. Our other adult children have had more than their share of disappointment and less than their share of success, despite academic brilliance. I am definitely in more physical pain when under a lot of emotional stress. I am sorry if this makes it sound like child related stress is, indeed unending. In my experience, it is.

That doesn't mean it is in your head, or that the stress causes the pain. I think it just creates a tension in the body which makes the nerves even more sensitive than usual. If the carpal tunnel is bilateral there is more to it. Did they check for pinched nerves at the elbow?

I went though a period of burning much as you describe. It faded away slowly over a period of weeks, leaving me only slightly more sensitive than I had been. It may clear up on its own.

I do not think anyone would say that pursuing a diagnosis is not important. It is, because many neuropathies are treatable. What is important is to come to terms with it if yours turns out not to be one of those. Generally speaking only hereditary neuropathies are completely untreatable, and yours does not sound like one of those since symptoms started in your hands.

Personally,I think a low fat diet is unwise, especially with nerve issues, and a few fish oil capsules are not going to make up the difference, but I am not an expert.

I do not know if neurontin will help with the pain you describe, it does help me with the burning and buzzing in my feet. It did not appear to affect the sunburned feeling. Nothing did.
I have no experience with Cymbalta.

If you have reason to believe stress is a contributing factor, would you be wiling to try meditation targeted to chronic pain, such as Jon Kabat-Zinn's Full Catastrophe Living? It is a little complex, but even simple meditation techniques can help you deal with stress better. Radical Acceptance is another good title. Good luck. What you are going through sounds very scary, but it cannot hurt to learn how to handle stress better while you look for a physical cause.