I would definitely recommend that you talk to a doctor
about this, especially since it has been such a chronic problem for you. You could talk to either your primary care doctor
or a dermatologist
The development of 'scaly' skin that you describe as 'reptile skin' actually has a medical name, which is ichthyosis. Although there are several severe genetic forms of ichthyosis, in most cases it is just related to the development of persistent eczema, or dryness and itchy inflammation of the skin that is made worse by cold weather as well as other things that irritate the skin (hot water, soap, chemicals).
Treatment for this condition does require aggressive moisturizing of the skin. It is best to use a thick, 'greasy' moisturizer (such as a product that contains petroleum jelly) as these tend to stick to the skin for a longer time and provider better protection. In addition to this, avoiding excessive showering and hot water can help.
However, most cases of persistent eczema do require, in addition to moisturizing, prescription steroid and other creams from your doctor. Therefore, start by scheduling an appointment with either your primary care doctor or your dermatologist, who should be able to give you some good advice.