It's typically helpful if you share screenshots of what you mean by things like "white parts appearing".
Given that, the following are just speculation on your problem:
Your confusion might stem from a few things you don't know Photoshop does automatically and some settings when you pack that you might not have noticed.
I encourage you to give this a quick read: Premultiply Alpha
Notice the screenshots, see if the patterns look familiar.
If that's the pattern you're seeing, and you get what "PMA" means, you should also know that just the act of (re)saving a PNG in Photoshop will cause it to apply Bleed.
A solution, if that's the case, is to just:
- pack your original texture with PMA off, and
- use Unity's
Sprites/Default
shader instead of Spine's default "Spine/Skeleton" shader.