There’s nothing wrong with your UVMap… The only problem, as I pointed out above, is that Normal maps are encoded into the UV layout. You cannot smear them, because they will sure break…
And as I illustrated, having normal maps that don’t show continuity (as a texture), doesn’t mean they are incorrect. They may well be continuous in tangent space, and that’s all that matters.
Another small example… If I tell you to ‘turn right’, in your space, you’ll turn right, but in my space you’ll turn left. At the edge between our spaces, there must be a conflict, even if for all that matters, you’ve turned into a fixed location in the world space.
The values for each pixel in a normal map, represent how much the normal should shift in the U, V, and W directions (which come from the UVmap itself). They live in their own space, and each will have their own ‘North’. You cannot treat them as if they are in world or object space.
Bottom line is: You shouldn’t ‘paint’ normal maps! They are mathematical entities, not some canvas with colors.