Important Notes

Drag-n-drop the ‘Sticky Textures’ modifier on to the object.
Make sure this modifier is above any ‘Armature’ or deformation modifier in the Modifiers tab.
In the Clay Doh material settings, under “Texture Mapping”, turn “Sticky Textures” to 1.

Right-click on your object and select Shade Smooth. This will fix “hard edges” and “split faces”.
When using ‘Curve > Clay’ modifier, select your curve and under the Curve Settings, set the ‘Fill Mode’ to None. When ‘Fill Mode’ is set to anything else, the Curve becomes a Mesh.
When using ‘VDB > Clay’ modifier, do not assign a material VDB object’s material slot. Instead, assign the material via the Geometry Nodes modifier.
When using ‘Mesh > Clay’ modifier, if there are any issues/ artifacts, please select all faces in Edit Mode, press Alt+N and select ‘Recalculate Outside’. This will fix any issues with face normals.