Your GLSL code shows that you use FP32, not FP64, so those artifacts are expected at LODs >= 15.
One solution is to use FP64 ( dvec3 ), however not all GPUs support all the operations you need for procedural noise.
In the I-Novae engine, I ended up using an hybrid solution: FP64 shader emulation. You can read more on it on this page:
For performance reasons many operations still work in FP32 ( only the inputs to the procedural function, ie. the planet position, is using double emulation ), but that alone significantly improves the precision on the lowest LOD levels. In my tests, I can go up to level 18 with that "trick".