Hi Nan, I can imagine how - literally - nerve-racking that can be!
Skin, nerves, inflammation, infections and/or an allergic reaction come to my mind first. And looking it up there can be about 20 causes (not Raynauld's): diseases, triggers, deficiencies. So these would definitely make it necessary to go to various doctors, to get all the following checked:
If it were "burning foot syndrome" it would be mainly in the soles (but also ankles & legs) often with tingling, at night, no redness, no tenderness, but sweating. That may be Small Fibre (Poly-)Neuropathy, SFN (= nerve damage). Whether or not it's that it may have to do with thyroid, diabetes, RA, kidneys, genetics, narrow blood vessels, meds incl. chemo, alcohol, heavy metals (arsenic, lead, mercury), or deficiencies (zinc, B-vitamins, esp. B5 or B12, B6, B1, or vit. E, copper, or iron (anemia).
(wikipedia, verywellhealth...)
In the meantime you might want to check med or supp side effects.