For those exploring other options, HAProxy has a nice balancing option, 'leastconn':
leastconn The server with the lowest number of connections receives the
connection. Round-robin is performed within groups of servers
of the same load to ensure that all servers will be used. Use
of this algorithm is recommended where very long sessions are
expected, such as LDAP, SQL, TSE, etc... but is not very well
suited for protocols using short sessions such as HTTP. This
algorithm is dynamic, which means that server weights may be
adjusted on the fly for slow starts for instance.
"This only applies when no persistence information is available, or when a connection is redispatched to another server", which means this only kicks in when no sticky session cookie is found.