A typical fix for this issue is to use an F5 OneConnect Profile with a single host (/32) mask (the mask is really important, see further reading if you’d like to know why). Of course, without knowing too much about about your overall config, I’d suggest to both validate this feature against the version of TMOS (Traffic Management OS - The F5 OS) the you are running and test this with a staging/test VIP first as this would be a change on the way the F5 device behaves.
By using OneConnect, the client is not fixed to a backend server by a TCP connection, rather it will load balance the HTTP requests individually so if in the same TCP session it sees different cookies with different persistence information, it will honour that. This also ensures the cookies are set with each HTTP response.
A similar issue on F5 Dev Central: https://devcentral.f5.com/s/question/0D51T00006i7i5Q/always-send-cookie-problems
And more detail here in these F5 KB’s: