Your dermatologist has the advantage of seeing your problem. I can only go by your brief history. However, I am a bit skeptical that you have "chicken skin" disease, which is usually another name for Keratosis Pilaris. KP is not supposed to occur in the axillary areas, but on the lateral portion of the upper arms, cheeks, and thighs.
Your description fits better with a rather rare entity called Fox-Fordyce Disease. This is essentially an apocrine miliaria (plugged up apocrine sweat pores). Apocrine glands are glands which release an entity which was once used to attract mates...think monkeys sniffing the butts of the opposite sex at the zoo. Usually the bumps affect all the hair follicles in a given area.
We do not know what causes Fox-Fordyce, but nearly all cases are found in females of the reproductive years so a hormonal cause is speculated. Fox-Fordyce is worsened in hot climates due to aggravation by sweating.
Treatment is not too good. You might try a mild form of Tretinoin such as Atralin or Renova. Surgery has also been tried in severe cases but I would not endorse this.