Is my prayer valid after touching a wet towel from an impure mattress, and I did not wash my hair that the towel touched?

If there is doubt about whether a wet, pure object met a najis (impure) spot, then the transfer of impurity is not ruled, because the default state is the persistence of purity. If you are certain that the wet towel touched the najis spot on the bedspread, then whatever it touched becomes najis. If you are certain that what touched your body and hair was the najis part of the towel, then you must purify them. Your prayer, in a state where you have not washed your hair, would be invalid and must be repeated. If there is any doubt about any of the foregoing, then there is no obligation upon you.