I don't really know how long it will take for you.
I felt quite a bit better by three/four months. It took me a year until my annoying "best friend" anxiety was gone, but still I handled everything better without alcohol. I consider anxiety a CNS issue. It took over two years until all my noticeable neuro symptoms disappeared completely. I didn't go to the doctor for any of them, I've had enough doctoring prior to getting sober and I really didn't want to take any pharmaceuticals to get through early sobriety, so I was on zero drugs. I'm still drug-free. (Well, except caffeine.)
I think staying sober is the best thing. The healing is gradual. You'll get better or you won't - but at least you'll be giving your body the best shot at health.