Hello Patricia (@pagray24), I have no medical training or background but I did a little searching and found references discussing adrenal insufficiency (Addison's Disease) that point out that is caused by an autoimmune disease. Here is a list of a few sites that I found with the information:

National Institute of Healh site:
https://www.niddk.nih.gov/health-information/endocrine-diseases/adrenal-insufficiency-addisons-disease
National Adrenal Diseases Foundation:
http://www.nadf.us/adrenal-diseases/addisons-disease/
Pituitary Network Association:
https://pituitary.org/knowledge-base/disorders/adrenal-insuffieciency-addison-s-disease.
Autoimmune health related problems can be difficult to pin down. I like that you are being your own advocate and searching for answers. I like to use Google Scholar (https://scholar.google.com/) when doing research mainly because it finds articles and research information/studies and allows you to show the most current links for information by selecting the year at the left side of the links shown.

I don't have the answer for your question on when to take hydrocort (Cortef) and Dexilant but I'm wondering if just a simple call to your local pharmacist may be able to help to see if they have a recommendation.

Hoping you find an answer soon.

John