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.

The reason you have mushrooms is you are watering too often. What we in the grass world have determined to be optimum watering schedule is for the lawn to get about an inch of water per week in the summer and an inch per month in the cooler months. That should be applied all at one time, like a rainstorm might do, rather than 1/7 inch applied every day. When you allow the soil to dry out between watering, you very seldom get mushrooms.

If you want to tune up your new patch of soil, I would suggest using organic fertilizer for the rest of the summer. Normal soil contains tens of thousands of species of fungi, bacteria, protozoa, amoeba, and microarthropods (teensy insects). These microbes usually perform many functions including keeping plants healthy, keeping each other healthy, cleaning up spilled stuff (including food stuffs and even chemicals), AND they provide food to plants through the roots. Without getting deep into the biology, when YOU feed the microbes with real food, they will return the favor by turning your lawn a very deep green, making it dense, and causing it to grow a little more. The food I"m talking about is organic fertilizer. The really good organic fertilizers are made from food like corn, wheat, animal parts, soybean meal, alfalfa, and any other ground up nut, bean, seed, and grain that makes sense. These healthy microbes self balance as long as they are fed and watered. Normal soil has a healthy balance of fungi, bacteria and all the rest. Diseases are suppressed naturally by the activity of the other microbes. When you do something like apply a fungicide, that wipes out a significant part of the balancing act. Other microbes will move in to fill the void until the fungi can repopulate.

The other thing I would suggest is a soil test to determine if you have any chemicals in the soil that might lead to issues with your dog. The best soil test for lawns in the states is probably the $20 test at Logan Labs in Ohio. If you get it tested and post your results here, you will get help interpreting the results.

All this is to help get your soil tuned up and rule out the soil and/or lawn as being an issue to the pup. It could be the dog is allergic to the grass itself. I know I am allergic to bermuda and to buffalo grass. I would not want to be rolling around in either one.