Hair Loss and bad eczema

My baby is 5 months old and I have been dropping hair all the time from the third month (sometimes in chunks 😱). I am also experiencing bad eczema outbreaks from the legs to hands (super itchy) which leads to scarring 😭.Doctor prescribed Aqua cream which is not helping. Anyone has similar bad hair loss and eczema problems? Any solutions? My baby is licking and sucking everywhere and pulling my hair nowadays so I need babysafe products to use and trying to prevent my hairs from falling all over my LO. 😝 Anyone can advise? Thank you.