Probably the tree needs to figure out where it's at. My new trees did the same thing. But the leafing out on one side is an indication of root damage. Not good!

My nectarine trees have never had fall color. They generally hold their leaves a lot longer than I'd expect (they outlast my pear trees), and they're still green when they finally drop them.

My peach tree leaves rarely change color. A few turn yellow but they are usually very small leaves. My five peach trees still have their green leaves on them and it is November (60 degrees this week in Newport, RI). This happened once three years ago (the temp.). I wail till they fall then spray copper. Mrs. G

Drew51, is that what causes the unilateral leafing out? Any chance this will ever be a healthy tree, or should I dig it up?

I'm not sure? Someone with more experience needs to comment. All i know is often the strange leafing is a result of root damage. Often it shows up on say where construction near a tree happened.
In your case one side of the roots were somehow damaged?
What I would do is email Stark Bros. about it, describe the problem, and say you fear root damage. They may send you a new tree next spring. I lost a new cherry tree from Grandpa's Nursery. It just died late in the season. Lost all it's leaves in August. they are sending a free replacement next spring..