How to prevent students from having an unfair advantage on exams, when they are held on different days?
There are various ways of preventing students from showing up on the wrong day or showing up more than once. The first step would just be to tell them straightforwardly that it's not allowed. You could also assign seats for the exam. Assigning seats has the added benefit of reducing opportunities for copying from a neighbor's paper; they can't just choose to sit next to a friend or sit next to someone they know is doing much better in the class.


  I teach a writing course [...] The classes are large, and the school offers me no assistance in managing the exams.


The enrollment of writing classes taught by a single instructor is usually limited because it's not practical for the instructor to grade large amounts of written work from a large number of students. At some schools, writing classes are large, but there are TAs who read papers. If you're a single instructor teaching a writing class to a large number of students, then that's a problem in and of itself; these issues with exam arrangements would then be a symptom of the more general problem.