It sounds like the problem is in the Load Balancer, as you yourself have said.
You need to change it to put a user towards 1 of your servers and keep them to that server.
That is out of the scope for this forums most likely as we don't know what load balancer you use.
The problem with the xpage app is most likely that the application uses session or app scope to keep track of where each user is.
If your load balancer use "round robin" style to spread every other request to each domino server you loose that app or session scope across the servers.