KarenArushi Thank you for sending us the repro project!
The texture of the skeleton that was exported with Premultiply Alpha seems to be enabled, but the texture settings in Unity were the settings for straight alpha textures, so you should fix them according to this section of the documentation: Advanced - Premultiplied vs Straight Alpha Import
However, for the issue of artifacts on the edges, it seems to be due to Write to Depth. Harald's answer in the following thread may be helpful: https://esotericsoftware.com/forum/d/17645-spine-2d-shaders-trouble-with-unity-visual-effects/3
When having Write to Depth enabled you will need to increase the Depth alpha cutoff value as it's currently set to the very low value of 0.001. Semi-transparency combined with depth-write will however always be a tradeoff between cutting off too much (higher Depth alpha cutoff) and cutting off too little (your current state, low Depth alpha cutoff value) while still setting the depth value with a semi-transparent pixel which mostly shows the darker background and blocks subsequent overlaid pixels.
In your case, it seemed to me that setting the Depth Alpha Cutoff and Shadow Alpha Cutoff to around 0.9-0.95 for the material using the Spine/Sprite/Pixel Lit shader would give good results. However, as Harald explained, there will be areas that will have too much cutoff.
Harald may be able to offer you better advice, but is currently on vacation. So please wait for a while for his response.
