May have nothing to do with the problem, but check soaps, body cleansers, laundry detergent, etc.

My problem was itching and scratching. The doctor recommended several changes, starting with Dove soap for sensitive skin, dispensing with all products such as Bounce in the drying cycle, and use of Tide-free as the detergent (and as noted above by several, stay away from perfumed products).
As a skin lotion, she recommended a product called Neutroderma for a start.
The skin specialist echoed the recommendations, but preferred another product for the lotion, but it was less effective than the Neutroderma, and very expensive as well. Also prescribed was Beniderm, a prescription in cream form, to use in severely affected spots. It does help, but can't be used for large areas (cortizone included in the formula).
I don't have scaling, so doubtfully your problem. My eyes are sensitive to dry air, but the common saline drops are sufficient for me.

The itching is a lot less, but I still have it. It is a common complaint here, where the air is very dry and cold in the winter, and often there is thin cloud cover, obscuring the sun (which is low in the winter anyway- central Alberta). The well-to do can spend a lot of winter time outside the province (Hawai'i, etc.).

It may take the allergist-dermatologist a while to narrow down the cause of your problem, but actions such as the above may help. My daughter-in-law is a physician, and she keeps me informed of new treatments.