connections without transactions continue to enter F5 and create a full session in the pool

Hi Margvin , 

Check your tcp profile which assigned to your virtual server ( in the client and server sides ) and check idle timeout intervals , you should set a suitable time ranges the default should be 300 sec == 5 mins for the connection to be idle. 

Please review this article too : https://my.f5.com/manage/s/article/K13004262

Also this an important article for idle sessions definations : https://my.f5.com/manage/s/article/K7606 
it's important to keep all objects and profiles with suitable idle timeout like ( TCP profiles , SNAT ) which I think you're using in your current deployment.