Peach tree disease
Your problem may be bacterial canker aka gummosis, or it could be something like peach tree borer, the damage of which can sometimes appear similar. Booth will cause an oozing of sap, though the gummossis can be traced to dark circular areas of damage on the trunk and stems, whereas borers can be traced to small individual holes.
Regardless of which is the problem in your case, control will be difficult and your main option, pruning out damaged branches, is the same. Obviously if the damage is on the trunk as well you won't be able to remove this, but often the damage starts in branches, so if you've caught it early enough, you can treat it by removing those parts. If it is borers, there are some additional options for control, outlined here: https://extension.arizona.edu/sites/extension.arizona.edu/files/resources/peachtreeborer.pdf
Here is another publication on gummosis: http://www2.ca.uky.edu/agcollege/plantpathology/ext_files/ppfshtml/ppfs-fr-t-8.pdf