often and using abrasive cleansers may cause irritation, tingling or burning sensation over the palm which may suggest a dermatitis. I suggest that you have this assessed by a dermatologist. You may need oral antihistamines for this.
With a history of asthma, then this itchiness may be secondary to a dermatitis specifically irritant or allergic type. You may need oral antihistamines to alleviate the itching
with a mild cleanser. Soak in warm water with menthol or camphor to cool your hands. Try to moisturize with vitamin E cream everyday .Avoid any known triggers.