Hello,
For this dry skin, instead of Neosporin, the mainstay of management is liberal use of moisturizers after washing with luke warm water. Soap should be used minimally
. Moisturizers should be reapplied liberally during the day. You can also use a topical cortisone (steroid) cream or ointment like dermacort(hydrocortisone) after consulting a dermatologist.
I sincerely hope that helps. Take care and please do keep me posted on how you are doing.