You said in your post that you are using a single Virtual Server and have multiple websites pointed to it.
We host multiple websites on a single domain hence single virtual server and single irule for all those websites.
If you are routing the traffic for the individual websites via an iRule, could you post the iRule that is doing the traffic routing so that we can see it?
Your second statement could be an indicator that there is something wrong in the iRule.
The issue we are seeing is when a consumer jumps a website LTM starts seeing two or more cookies and send the request to the wrong pool:node pair.
Nimbostratus