Would be much simpler to not model the strings but texture those patches instead. Could also make the geometry transparent in the material/shader where one should be able to see through it.
It’s great that you included an example .blend, don’t see that often from a new forum user. If you meant to include the background images, those weren’t packed in. Tutorial linked in my signature shows how to prepare a .blend for upload.
The objects in the file contained common beginner mistakes
Unapplied object scale and rotation. These make what you see in the viewport different from what tools and modifiers use, so they seem to behave differently from what you’re looking at. For example, you’re not looking at the actual thickness of what solidify creates.
Object mode, ctrl+A -> rotation & scale to apply.
Vertices are away from the mirror plane so they don’t merge. Could enable clipping on the mirror and move the vertices along X until they stick on the mirror plane and merge. Also don’t enable subdivision surface visibility on the editing cage while building the model structure, it doesn’t show the actual vertex positions and you can create problems such as this one without noticing.