Hi there! My name is XXXXX XXXXX I would be happy to help you with your question about Biscuit. I'm sorry it took someone so long to get back with you. Did you still need help?
I'm sorry to hear that Biscuit has been having some trouble with sores on his body and a cough! From what you describe, it sounds like he may have had an issue with food allergies. Cats can be allergic to ingredients in their food, at it may be that he was sensitive to something in the dry food that wasn't present in the wet food.
The sores and coughing may be a sign of the allergic reaction that he had to the food, or possibly even something he came into contact with, or inhaled (cats can experience inhalation and contact allergies as well, such as to pollens, molds, fungus, smoke, etc.).
Since he is still experiencing the sores and the cough after the food change, there could be a couple of things going on. It's possible that he could have a secondary bacterial infection that is keeping his skin from healing without antibiotics. Similarly, if the food switch was recent, it may take a while for his skin to clear up (sometimes it can take up to 8-12 weeks for the symptoms of a food allergy to disappear after a switch to a new diet).
Cats can suffer from coughing due to asthema, which can be brought on by allergies as well.
However, since he was also experiencing weight loss and a poor coat quality as well as a cough, it's possible that he might be suffering from an endocrine disorder. There is a disease older cats can develop called "hyperthyroidism" in which their thyroid gland starts overproducing the hormone that regulates their metabolism.
Some of the symptoms you might see related to this disease are hair loss and skin problems, poor coat quality, a sudden loss of weight, even though their diet or appetite hasn't changed, or even heart issues which can cause them difficulty breathing.
Since Biscuit is still having trouble with his skin and his cough, it would be a good idea to get him into a vet to assess his skin and determine if he requires an antibiotic, or if he needs to have some blood work done to rule out the possibility of another disease, like hyperthyroidism. If he is indeed having problems with allergies, your vet will be able to make some sugestions about medications to help with his allergies (one of the more common ones is over-the-counter plain Benadryl at a dose of 0.25-0.5 mgs per pound of body weight every 12-24 hours to help with symptoms of scratching, watery eyes and sneezing that can be associated with allergies).
Hyperthyroidism can be regulated through medication, but it can cause problems, and even become deadly if left untreated.
Since Biscuit's cough indicates that he is having a problem with his respiratory system, even if it is mild at this point, keep an eye out for signs that he is worsening. If he starts to breathe with his mouth open, if he pants, if he's breathing fast and heavily, if he becomes listless, or if his gums change from a nice pink to a blue color, this would be considered an emergency, and even something like asthma associated with allergies can become an emergency if any of the above signs develop.
If you have any follow-up questions or concerns, or if you need clarification on anything I have said, please feel free to reply to me until you are satisfied, and I would be glad to address any concerns or clarify any part of my answer. Otherwise, I hope this information was helpful to both you and Biscuit, and that he's on the road to recovery soon!