The shape is what I want but I need the star to appear on the surface, but SR keep placing halfway inside the object and playing with offset makes it float above it.
Tried solidify but it added too much thickness to it
It looks like your star shape doesn't have enough verticies to project onto that surface. Try add subdiv modifier to give the object more verts, this will help give it enough detail to project to the surface. You'll also want to adjust Shrinkwrap settings, especially "Offset" and "Snap Mode" will be helpful.
As also mentioned in the comment to your question, you could also consider adding that star as a texture. Here's a good explanation on how add an image texture to an existing material.