Please tell me how to care for dry, itchy skin.

My face started itching a few days ago.
It was itchy, so I kept touching it, scratching and rubbing it lightly, and as a result, it became slightly raised, rough, and red.
The itching gets especially bad after washing my face, so I keep touching it.
I have a lot of dead skin cells, so my makeup looks patchy.
I tried applying Centella Asiatica extract toner, but it didn't have much effect.
Even after reapplying moisturizer, the relief is only temporary; the inner dryness does not easily subside, and the itching persists.
I applied some skin ointment that I was prescribed a while ago because it was itchy, but it didn't seem to have much effect.
I would like to know a way to hydrate and soothe itching without irritation.