Hello Liz, 

This looks consistent with bacterial leaf spot for peaches. Xanthomonas bacteria will infect leaves, causing tan lesions that will eventually drop out of the leaves. Infected leaves can turn yellow and dieback. This could also potentially be coryneum blight, but these would both have the same control recommendations. 

The sticky substance is gummosis! This is common in stressed out stone fruit. It is a sign that you have disease, but not directly caused by the pathogen. 

I would spray copper on the tree this fall when the leaves drop and in the spring right as the dormant buds start to become active. Beyond that, pruning to remove diseased tissue is a great way to reduce the inoculum load. Peaches are fairly susceptible to bacterial leaf spot and frequently develop cankers caused by the same pathogen. Look out for sunken dead tissue with gummosis nearby and try to prune it out!

This pathogen is going to be most active in the spring and fall, so I would not necessarily worry about spraying this summer. Copper can potentially damage the leaves and cause phytotoxicity if sprayed during hot weather. If you think the pathogen is continuing to spread, you can spray copper this summer in cooler weather (<75F). 

 Peaches can keep living even with bacterial canker for many years, so don't worry too much!

Best,