[anonymously]

Environmental allergies wax and wane. Maybe this is why the dog was given up?
If the dog is suffering I would advocate for the dog, ask the vet to recommend to the shelter to have him seen by a specialist (as described in my previous posts). In my opinion, it will be cheaper in the long run. The best part is, the dog may feel and look better and therefore become adoptable.

Why would he want to walk and move around if he is bothered by pruritus.
You could try a dome dish to slow down his eating, the rescue may have one you could borrow.

Above comment is in response to:
“Regarding the allergy testing. I don’t think the rescue is going to do that at this point unless it’s recommended by the vet. His allergies are not severe I would say (not like our late dog who had terrible allergies). We just notice a bit of paw chewing/licking.”
And:
“He had a severe yeast infection and lost a lot of hair and was severely matted and had to be shaved down. His hair is growing back now and we’re giving him some fish oil supplements as well. He has been on occasion chewing or licking his feet/paws. I think that might be our grass and we’ve had another dog who was allergic to the grass and I can attest that it is very itchy for me to walk in it (I have grass allergies).”