You are just guessing about the cause and are making flawed assumptions. Yeast infections are common. They happen in warm, moist places. They have nothing to do with mushrooms. The mushrooms growing in your lawn are the fruiting part of a fungus that eats decomposing wood/roots etc. That fungus cannot live on animal skin. There is an outside chance that your dog is actually allergic to mushrooms. That would show up if he ate the mushrooms. I guess it is possible that he is having a contact reaction to them, but the chances that he has such severe allergies that way to mushrooms is exceedingly small.

Since you have just moved into a new housing development, chances are, your dog is reacting to something in that new environment. New carpets? New cleaners? Any chemicals (fertilizers, herbicides, insecticides) sprayed on the lawn? New diet? All it takes is a little irritation, then the dog scratches, then the skin gets raw, then infections can set it. You might be able to find it, but it is awfully hard since the dog can't tell you.

Yeast likes warm and moist, so anything you can do to minimize that will help. Dry your dog off after outings. Clean any carpets or bedding the dog lays on. They sell special shampoos to treat the current infection, so you probably want to give that a try.