Hello and hope you are doing well.
The symptoms described by you could be due a skin disorder called eczema. This itches so much that all the scratching makes it look like 'shoe leather' (or 'elephant hide'). People who suffer from this disorder should avoid contact with triggering factors like soaps, cosmetics, jewelry, clothing, and detergents. Sometimes sweat, changes in temperature and psychological stress are known to trigger these episodes. So, the best way to prevent attacks is to identify those substances which you are allergic to and avoid them.
So, Allergy testing would be worth looking into. Apply constantly plenty of good moisturizing creams. Use a humidifier in the rooms to avoid dryness.
Hope this helped and do keep us posted.
It's quite possible that you are having an allergic reaction to the minocycline.
Symptoms of an allergic reaction include rash, itching, swelling,...
As for the eczema...it is probably the source of your itchiness.
Perioral dermatitis:
It is commonly caused by the use of fluorinated topical glucocorticoid steroids to this area, although it can have many other causes, including fluoride toothpastes or simply continual irritation. It is exacerbated by cosmetics, soaps, facial washes and steroid creams so use of these things on the affected area is not advisable.
Licking your lips constantly can cause this as well...and over use of moisturizers, water, etc.