Sad to report that the upgrade to OpenSSL (v1.0.2.r) and recompiling of the nsopenssl driver did not resolve our issues. After testing it on our development environment we did the upgrade in Production with the same results.
Initially we only updated the software without enabling the site-wide configuration. We ran it for a day and our HTTPS-restricted applications showed no crippling problems but did notice some connections would be in a CLOSE_WAIT state for some time and would eventually go away.
This morning we forced HTTPS site-wide, and as the activity on the site increased we started to see intermittent delays until the site became unresponsive. netstat showed a large number of connections in a CLOSE_WAIT state. After some time, we received a "fatal signal 11" error and the site crashed. The crash is not something we had experienced before. We suspect this is because we had not left it running in this state for as long as we did today.
After rolling back the site-wide configuration, we restarted the service and noticed that HTTPS connections would still hang from time to time. In our original configuration non HTTPS-restricted URLs are redirected back to HTTP and these redirects would also take a long time. This is something we had not experienced before. It took restarting the server to alleviate this condition.
At this point, we have ran out of ideas on what to look for and troubleshoot. We are going to start testing the NGINX solution and hope we won't run into similar issues. I welcome any other tips or troubleshooting ideas in the meantime.