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The oozing is normally indicative of peach tree borers. Here is more information: https://www.gardeningknowhow.com/edible/fruits/peach/peach-tree-borer.htm This will help with brown rot: https://www.gardeningknowhow.com/edible/fruits/peach/brown-rot-fungus.htm

Dormant oil and sulfur spray are essentially used for the same thing, so you can choose one or the other to apply. To keep brown rot at bay, start treating the tree in early spring before the blossoms open and reapply the fungicide every 2-3 weeks while the tree is in blossom. Reapply the fungicide again when the fruit starts to ripen and changes color, which is normally about 3 weeks before harvest. Here is more information: https://www.gardeningknowhow.com/edible/fruits/peach/brown-rot-fungus.htm

Technically, peach and nectarine are the same fruit. There is literally only one gene difference between the two (which results in a fuzzy or non-fuzzy fruit). Beyond this, cross pollination only affects the seeds (the fruit of tree planted from the seed would be affected). It is only something you need to worry about if you plan on growing trees from the peach or nectarine seeds.

If you are concerned with insect pests, then treat the plants with neem oil. This is an effective and safe insecticide which also doubles as a fungicide. Here is more info: https://www.gardeningknowhow.com/problems/pests/neem-oil-uses.htm

I recommend neem oil. It is organic and very effective. It is often sold as fruit tree spray at stores. Check the label. Here is more information on neem oil: https://www.gardeningknowhow.com/problems/pests/neem-oil-uses.htm

This article should help: https://www.gardeningknowhow.com/edible/fruits/peach/peach-tree-care.htm