I don’t have a specific answer to the sock question, but I have a suggestion for lessening the burning. If you don’t already have this, ask your neurologist for some compounded pain cream. It is made at a compounding pharmacy. I get mine in the mail from Birmingham Al because that’s cheaper than my local compounding pharmacy.
It has lidocaine, gabapenten, amitriptylline and some anti inflammatory meds in it. If your neurologist isn’t familiar with it, you can find a compounding pharmacy and ask them what “recipe” they use. Different medications can be dropped or added. You have to pay out of pocket, but who cares at this point, right?
Hopefully it will help the burning.