Steroids can trigger the thyroid or the adrenal glands into improper function if they were on the edge of showing disease. The influx of steroids can put the natural steroid production out of balance. That might be why you are seeing her feel unwell when the steroids are skipped.
Standard bloodwork does not usually check thyroid or adrenal gland function but ask your vet about that being included in the panel run.
You may have started with a skin infection, had it go systemic, or the dog could have transferred the infection from one site to another via the licking, and now be dealing not only with an infection but an imbalance as the result of steroid use too.
here is some info on adrenal gland issues
http://marvistavet.com/html/cushing_s_disease.html
and here on thyroid issues
http://marvistavet.net/html/body_hypothyroidism.html
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