Hi and welcome to Spine!
The above information still holds true - as long as all active attachments of the skeleton are using the same atlas texture, it will work with the single-material limitation. So if you switch between skins that all have a separate single texture, it will work. However, note that if you programmatically change an attachment at runtime so that one of second atlas texture is used, it will cause the same problems.
Regarding the export:
One important question: have you set the atlas extension to .atlas.txt? If it is the default .atlas, it will not be read by Unity.
It is potentially easiest to export the skeleton at once, with textures packed with setting Image Folder:
Prerequisites: Ensure there is a folder for each atlas texture that shall be created and that the attachment images are placed in the desired folders.
Step 1: Press Ctrl+E or in the dropdown menu select Export...
Step 2: Enable Texture Atlas Pack and select Image Folder instead of Attachments right to it.
Step 3: (optional) Check in Pack Settings if under Options in the bottom right Flatten Paths and Combine Subdirectories is disabled (this is the default).
Step 4: Click Export.
I have now added a section to the documentation pages here:
spine-unity Runtime Documentation: Advanced Export for SkeletonGraphic