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You have azalea lace bugs. They will eat the leaves of azalea plants. This article from University of Florida will help you with them: http://entomology.ifas.ufl.edu/creatures/orn/shrubs/azalea_lace_bug.htm

Several things could cause this, including a lack in phosphorus. This would cause the plant to stop blooming. Add some phosphorus to the soil. This article will help: https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/phosphorus-plant-growth.htm It could also be due to a microclimate in your yard. This can be caused by hills, buildings, dips in the ground and even other plantings. Microclimates can cause an area in a yard to be up to a full zone higher or lower than other areas of the yard and may be as small as just a few feet.

It sounds as if your azalea has rust. More information on rust can be found here:
https://www.gardeningknowhow.com/plant-problems/disease/learn-about-plant-rust-disease-and-rust-treatment.htm

To keep the rust under control, you are going to want to remove and dispose the affected leaves and treat your plant with a fungicide, such as neem oil. Continue to remove leaves and treat the plant until all signs of the plant rust are gone. Furthermore, be very conscientious about watering - rust thrives in a wet environment so you will want to be careful not to overwater your plants.

I would say that its a moisture issue. It does not look like a fungus and I do not see any pests. The leaves next to the brown ones look quite healthy. Up your watering and cut back the dead branches. Once they are back to healthy go ahead and fertilize in the fall.