Hi. I'm Dr. Gotthelf. I have been a skin and ear vet for 35 years and I would like to use my experience to help you with your pet's medical problem.
I seriously doubt that this is a food allergy. Panting heavily usually indicates that the dog is hot and is trying to blow off heat from his lungs and tongue. If he is having breathing problems, then a chest X-ray may be needed. A dripp nose, especially if it is clear may indicate an overproduction of tears in the eyes or an upper respiratory infection. None of these symptoms are consistent with a food allergy which shows up in the dog as either a skin problem or a diarrhea type of disease.
He is salivating as well. i guess this comes with panting?
Yes. You can take his rectal temperature and see if it is above 102.5 F. A fever can cause panting and a runny nose.
He sometimes feels hot in places. But it is ongoing. Is it possible that he gets fever often?