Your dog might have an infection causing the lymph node swelling or he might with the steroids being added in have an issue with an autoimmune disease like juvenile cellulitis also called puppy strangles.
The steroids can increase appetite you are right about that.
You should call your vet and ask about the lethargy as it might be a reaction to one of the meds or just to being ill. You should find out how soon the vet expects to see a change for the better with the medicine regimen. That way you will know when to call if its not working.
Poor guy obviously doesn't feel well so if this treatment he is on doesn't help the problem you should consider biopsy of the inflamed nodes.
Lymph nodes swell in response to immune system demands and allergies or infection can make them enlarge.
Hope this helps you!
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