How do I clean a car headliner without making it sag?

Use as little moisture as possible. Headliner fabric is glued to a backing board, and soaking it can loosen the adhesive and cause sagging. Lightly mist a suitable fabric cleaner onto a soft microfiber towel or brush, dab or gently agitate the stain, then blot with a dry towel. Avoid scrubbing, steam directly on the fabric and saturating the material.