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I personally like neem oil as it is both a pesticide and fungus but does not hurt bees or other beneficial insects and is also safe for people. Here is more information on it:
https://www.gardeningknowhow.com/problems/pests/neem-oil-uses.htm

Peach Trees are know to be affected by numerous insect issues and diseases.

Borers are a likely problem due to the symptoms your tree is presenting along with several other fungal or fruit tree diseases.

The tree is likely partially girdled, thus the slow die back.

Fungicides can be used, but they are not as affective after the tree is all ready showing signs of death.

You are right to remove and dispose of all the infected leaves and limbs.

Here are some links for you that may offer some help.

https://www.gardeningknowhow.com/edible/fruits/peach/peach-tree-borer.htm
https://www.gardeningknowhow.com/edible/fruits/peach/common-peach-diseases.htm
http://www.clemson.edu/extension/hgic/pests/plant_pests/veg_fruit/hgic2209.html

This could have started out as a water issue. The heavy rains combined with the heat, these can cause fungus issues.

I would spray with Neem Oil, as do so weekly. It works as both an insecticide and fungicide.
Remove any dead material and dispose of.

I don't think is is a deficiency either but follow the fertilizing schedule in the links below.

https://www.gardeningknowhow.com/edible/fruits/peach/fertilizing-peach-trees.htm
https://www.gardeningknowhow.com/edible/fruits/peach/peach-tree-care.htm
https://www.gardeningknowhow.com/edible/fruits/peach/common-peach-diseases.htm

Depending on the color and nature of the mold, it could be powdery mildew or sooty mold. Both are fungal issues with one brought on through excessive moisture/humidity and the other by insects pests like aphids. Neem oil can take care of both problems - the fungus and the pests. Here is more information: https://www.gardeningknowhow.com/problems/pests/neem-oil-uses.htm

Not knowing the variety or your zone, it's entirely possible your tree is getting to the end of it's life.

Some varieties produce fruit till they are 15 to 20 years old.
Cooler climates can shorten the life of a Peach Tree, where in warmer climates with great growing conditions a Peach Tree can live 50 years.

The loss of a limb can be a stress to the plant or a symptom of other problems with the tree.
Check for signs of insect or fungus issues.

I'm listing some links to refresh you on the care requirements.

https://www.gardeningknowhow.com/edible/fruits/peach/peach-tree-care.htm
https://www.gardeningknowhow.com/edible/fruits/peach/tree-with-no-peaches.htm
https://www.gardeningknowhow.com/edible/fruits/peach/peach-tree-borer.htm
https://www.gardeningknowhow.com/edible/fruits/peach/common-peach-diseases.htm

You can store the dry seed in a plastic bag in your refrigerator provided you do not put apples or bananas in the same fridge (because they release a ripening gas). Alternatively, you can store the pits at room temperature, but out of the sun.