Okay, start by using the search button here: /forums/search/allergies/
It sounds like environmental allergies (at the least), may very well be why the dog was given up.
I will be straight with you, the dog needs to go to a veterinary dermatologist and it will probably cost about $1000 for a diagnosis and to start treatment. Maybe a few hundred a year to keep him stable.
If you do not have pet insurance, or, cannot take on this kind of an endeavor, talk to the shelter. Otherwise, if you want to keep him, I would make an appointment with an Internal Medicine Specialist and go from there.
Regarding the food, you could try a limited ingredient grain free kibble, such as Nutrisca salmon and chickpea (check chewy dot com) see if that helps. Add a little water to the kibble.
