I have the exact same problem and at times I am in utter despair. While I agree with many of the suggestions, it’s very hard to focus on nutrition, exercise and mindfulness when you are literally paralyzed by brain fog. You analogize it to an “itch,” for me it’s literally paralysis.
I have had success taking a combination of ADD medication and natural amino acids (tyrosine and 5-HTP combined in a product called NeuroReplete). My functional medicine doctor advises me on the dosage for the amino acids with the knowledge that I’m taking adderall. The I understand it, stimulants like adderall improve the brain’s use of dopamine but can deplete dopamine stores. The amino acids help with production. I’m not a doctor and do have not verified the veracity of that statement, but I do find that my brain fog tends to taper off after about a week of adding in the amino acids. Long term, the best solution seems to be nutrition, sleep, exercise and stress reduction, but I need a focused brain get me started.