Try cutting out night shades (potatoes, tomatos, pepers) from your diet and see if that helps... They are a gut irritant. Considering you have so few things in your diet, potatoes could be a significant irritant.
Also try eating an avocado on an empty stomach - does that give you heartburn? Some people with sensitive digestive tracks do not digest avocado well (myself included).
I know you didn't mention fruit but make sure to eat no at all fruit. The fructose/sugar from fruit promotes Candida and will increase itchiness. I have excema myself and my ND had me cut out all fruit & nightshades and I've noticed a worlds of difference.
Also try to expand your diet with a variety of vegetables/ greens; kale, brussel sprouts, swiss chard, bok choy, mustard greens, cauliflower, asparagus, celery, fennel, zucchini, snap peas.. It's exceedingly important to vary your diet because your body can begin to develop intolerance's if you eat too much of certain foods, especially if you have a sensitive digestive tract..
In regards to your face/skin.. try coconut oil on your body, and if you don't want to use it on your face, try jojoba oil - it mimics the skins natural oils and is recorded as non-comedogenic (coconut oil is said to be comedogenic, but many people would disagree from their own experience).. Aloe Vera is also calming.
In the mean time if you need a supplement to help with the itchiness stay away from over the counter anti-histamines... An excellent natural anti-histamine is Quercetin
it is available at whole foods and most natural food stores. it is even safe to take 3 pills at a time if you're having a significant out break. this will help while you're diagnosing the root cause, try not to rely on it if you don't have too - but I know it will make life far easier (helps people not waking up itching)