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There are several insects that can cause deformed and oozing fruit. This article has information: http://www.clemson.edu/extension/county/laurens/mg/askmg/076_peach_tree_worms.pdf
Also there are fungal diseases that might cause this. To correctly diagnose the problem, take some fruit samples to the Extension Service. This link will help you locate the nearest one: https://www.gardeningknowhow.com/extension-search/

Your tree should not have died on the way home, unless you drove through a deep freeze or car wash, or some such thing. Or you left it sitting in the vehicle after you got home for several days so that it dried out. However it happened, the dead leaves are gone. If the branches are still bendy and green, it might make new leaves. If they are crispy and snap when you bend them, they're all dead. The green leaves around the bottom are leaves from the root stock. Fruit trees are typically grafted, that is a tasty fruit variety of plant is grafted onto a root stock of a tougher but less tasty variety of the same specie. Growth from the root stock is pruned off so the tasty fruit gets all the energy -- unless the tasty tree is dead. If you know you didn't do anything to kill the tree, you might try returning it to the store. For future information, here are some article about planting and growing peaches: https://www.gardeningknowhow.com/edible/fruits/peach/peach-tree-care.htm
https://www.gardeningknowhow.com/edible/fruits/peach/fertilizing-peach-trees.htm
https://www.gardeningknowhow.com/edible/fruits/peach/peach-tree-thinning.htm

Sounds like brown rot, a very common infection. Learn what you need here: https://www.gardeningknowhow.com/edible/fruits/peach/brown-rot-fungus.htm

One question -- are the balls still hard when they fall off the tree? Here's an article that tells you more about the possible causes of peaches falling off before they ripen: https://www.gardeningknowhow.com/edible/fruits/peach/peach-tree-fruit-drop.htm

The application of Green Cure fungicide is earth friendly and about as safe a treatment as you can get. Here is a link to their website, check it out: http://www.greencure.net/
I have used this product on my rosebushes and on my cucumber and pumpkin crops successfully for several years. If you have another product of choice you wish to use, read the label on the product well to see what it says about spraying while still bearing fruit, or call their 800 line if the label information is not clear.

Yes, the hard winter has affected trees and plants around the country. Another cause might be age -- peach trees need a few years to really get going. Here are a couple of articles that may suggest avenues of investigation you can use to understand the problems: https://www.gardeningknowhow.com/edible/fruits/peach/peach-tree-not-leafing-out.htm
https://www.gardeningknowhow.com/edible/fruits/peach/tree-with-no-peaches.htm