Answer: It sounds like a disease; a fungal disease called gummosis is the likely culprit, but it could be a bacterial disease. Since I can't diagnose it via the internet I strongly recommend you take samples of the leaves and a photo of the base of the tree to your local Cooperative Extension office or Master Gardener Plant Clinic. Helpful folks there can diagnosis the problem and make treatment recommendations.
Best wishes with your peach tree!
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