There are two different things causing these issues, both of which needs to be dealt with if you want a good result.
Best way to do this is to add some padding to your textures as described in the blog post you linked to.
This will remove (or at least reduce by pushing away) the bleeding that causes lines.
Second thing is to remove artifacts caused by the Frac node.
I recently started a thread about this hereand was able to help me completely remove all the artifacts.
Just let me know if you have any questions.
Edit beat me to it 