Do your generic 3D print prep work
UV-Unwrap the object.
It's best to use a manual unwrapping method (mark edges) as the smart UV unwrap leaves visible marks on the texture. (Although this might be solved when increasing the gap between islands)
Make sure to have large gaps between islands (can a safe minimum be calculated?)
Problem with visible seams in 3DPrint: is it because it's texture downsampling option?
Open the texture and lay out the UV (or use Cycles to bake the 3D view to the texture)
Save as a PNG file, so it's external to the project.
connect your texture to the Principaled BSDF shader, and the shader to the material surface output. This might look bad, but otherwise the texture is not linked in the exported .mtl file.
Export as OBJ & set Path type: copy (blender copies texture file to destination folder)