HTTP Domain Cookie not working on new 11.3 version

Your iRule is good. On a v11 system with partitions, the [LB::server pool] command will return the partition and pool.

/Common/local-pool

You just need to make a minor change:

when LB_SELECTED { 
     get the pool name used for this request   
    set my_pool [string map {"/Common/" ""} [LB::server pool]]
    log local0. "my_pool $my_pool" 
} 
when HTTP_RESPONSE {
     Check if the persistence cookie exists in the response 
    if { [HTTP::cookie exists "BIGipServer$my_pool"] } { 
         set the domain attribute on the persistence cookie
        HTTP::cookie domain "BIGipServer$my_pool" "foo.com" 
    }   
    if { [HTTP::cookie exists "JSESSIONID"] } {
        HTTP::cookie domain "JSESSIONID" "foo.com" 
    }    
}