Have you tried fish oil? This often helps improving skin & coat condition.
When I got my boy from the shelter (now over 5 years ago) he had quite a few issues too, digestion issues, hotspots and obsessive itching/licking to the point of bald raw broken patches. I put the itching down to nervous/stress behaviour as there was no evidence of parasites, or any change of symptoms by tweaking his diet to indicate an allergy. A raw diet, supplements, topical use of cardiospermum tincture ('nature's cortisone') and a stable loving environment sorted him out eventually, but it was a long, slow process, it took 6 months or so.
If he's only been with you a few weeks I would allow more time for him to settle down into his new routine and good food regime, and maybe try a few different things before resorting to long term use of strong drugs.
Good luck!
