best way to deal with load balancing web services where users' IPs are changing?

What's the current best practice for load balaning web services in general these days? My current old school Cisco load balancer will serve a given user from a given server for 10 minutes, but it determines what a session is by the source IP. In a world of mobile devices this is probably pretty common? This causes issues in some of our web apps because our load balancer doesn't retain their sticky session to a given server and they can bounce which servers they are on in the backend... in turn that causes them to land on different databases.

Is this an application architecture issue to solve? (I can think of ways to do so... but not great ways). Or are there more modern load balancing approaches that can determine what a session is in some heuristic manner other than IP address alone--so if their underlying IP changes their session remains sticky still?