Oh, your poor boy! It's so frustrating to see them dealing with problems that take so many appointments and so many tests to figure out. I hope testing a new cleaning system is enough to remove the irritants; it's also a faster and more convenient course of action than waiting for vet appointments and test results.Here are two photos of the paw issue. The main paw pad one is not as clear of a photo. The problem starts as almost like a rash on the edge of the main paw pad and then gets dark bacteria around it, ulcers, and eventually scabs. The problem then moves to the edges of the toe pads on the same paw.
Two things you can ask your vet about in the meantime are fish oil and epsom salts. Neither will be a miracle cure right now, his allergies seem too intense at the moment, but they can help. The fish oil will help his body's inflammatory response to the allergens; one of my cats has major dust and mold allergies and fish oil helps tremendously to controlling her symptoms. And while an epsom salt bath or wrap may sound odd, it can provide a lot of relief for swelling and itching. A cat in our rescue needed this after a severe case of dermatitis and it helped a lot.
Definitely clean everything his paws come into contact with, since he could be spreading the irritants as he walks around. So flooring, bedding, litter boxes, any favorite sleeping spots, that sort of thing.
Another route to explore could be environmental allergies related to furniture and belongings. It's possible some belongings brought old allergens into a new home, or exposure to new furniture could cause something like "sofa dermatitis". But I think cleaning products are the right place to start for now. Whatever the cause, I hope you find relief very soon!