I agree with Drew from what info we are working with.

If it become colder than -18F the wood can become damaged. It sort of looks like that, hard to know, don't have a clue where you are. What kind of peach, how old, etc. Need as much info as possible.

We are located in Dallas Texas. So our winters were pretty mild. The tree was here already when we moved in a year ago so I am unsure what kind of peach tree and how old it is. But my guess would be at least 4 years old and a 7 inch trunk size. We have a slightly smaller peach tree near it that is fully leaf'd. Last year, with the bigger peach tree, it had a lot of peaches but most of them mummified/curled up. Some of the bare branches do have a few leaves on them.

We also noticed a clear hard substance coming out of some of the peaches.

Unsure about the branches, I don' have enough experience. The problems with the mummified peaches is probably a fungal infection. All peach trees have to be sprayed a few times a year at least with fungicides. The clear substance is probably from an oriental fruit moth infestation. Or another insect. Insecticides have to be sprayed on peaches on a regular basis.

I would take old tree out and use new tree. In Dallas need peach tree flower very late set fruit, because late freezes kill little seed then rot inside peach looks good on outside, but early ripen start peaches turn brown all peaches on tree you can't spray this type rot. Going be tough growing peaches in Dallas after late freezes over short time comes high temperatures.

You might have Bacterial Canker disease that has girdled some branches at some locations. The branch will die from the girdled position outwards. I lose branches like that occasionally.

First of all, that tree is in need of some very severe pruning, and is way
too close to the fence. Both of which make it very hard to manage.
If it were me, I'd get rid of it, and start over with a variety that I liked and
I knew would do well in your area. Peach trees are fast growers, and with
the proper tree, planted as a whip this coming winter and properly managed, you should have an abundance of quality peaches within 2-3 years.