Persistent cookie value changing on each reload of page

Hi n8all3n , 
Please check your Cookie Persistence profile : 

open ( Local traffic >> Profiles >> Persistence >> then select your cookie profile). 
you will find this Feature " Always Send Cookie " , most properly you enable it and this make bigip update the persistence cookie with each response , so disable or remove this check box , delete all connections for this Virtual server , then test it. 
After that bigip should send only one persistence cookie with the first response and use it each time for persistece. 

For more info , Review this https://my.f5.com/manage/s/article/K83419154 , specially " Always Send Cookie " Part.