汚れる and 汚す: a stain and its cause
Separate the state of dirty clothing from what made it dirty, and practice negative requests and expressions of degree.
The state left on clothing
雨の日に歩いて、靴が汚れました。 says that the shoes got dirty from walking on a rainy day. 汚れる is read よごれる and describes the change in the shoes.
床が汚れています。 can report that the floor is dirty. It does not by itself identify who made it dirty or whether anyone is cleaning it now.
Name what was made dirty
インクで机を汚してしまいました。 means that the speaker got ink on the desk. 汚す, read よごす, marks the affected object with を and can mark the material causing the stain with で.
To ask someone not to get a wall dirty, say 壁を汚さないでください。 汚す has the negative form 汚さない; 汚れない instead describes something not becoming dirty.
汚い and expressions of change
このタオルは汚いです。 evaluates the towel as dirty, while タオルが汚れました。 reports that it became dirty. The focus differs even when both concern its present condition.
袖が少し汚れたので、そこだけ洗いました。 says that the sleeve got slightly dirty, so only that part was washed. 少し limits the degree of dirt, while そこだけ limits the area washed.
Which asks someone not to dirty a desk: 汚さないで or 汚れないで?
Show the answer and why
Say 机を汚さないでください. Use the negative of 汚す because you are asking the person not to act in a way that dirties the desk.