The dry skin
you are mentioning may be some sort of dermatitis. Many times it mainly involve the hand only. Hand dermatitis can be because of some sort of allergy
to detergent or soap. So if you find that some specific detergent or soap or any other allergen is causing this problem than, the best thing is to avoid this allergen.
The medication which you are prescribed are also to relieve the allergic dermatitis
and this medication also works for you. So most probable, it is some sort of allergic dermatitis and require care to prevent you from the allergen.
I would suggest you to consult some dermatologist
and try to use low allergic detergent and soap and try to keep your hand moist and clean.
Thanks and take care.