George: from the description of the behavior I'd guess that there's a proxy upstream of the BigIP. Have a look at this solution if this is the case: http://support.f5.com/kb/en-us/solutions/public/7000/900/sol7964.html?sr=11057641
The basic idea is that you enable a oneconnect profile (use a /32 mask: 255.255.255.255), which will enforce a detach and a layer 7 evaluation for every request. Then your cookie persistence will start working again. This may not be the issue, but it sure sounds like it. I've run into this several times and oneconnect usually fixes it outright.
Good luck!
-Matt
Nimbostratus