RHEL: Possible to upgrade OpenSSL version of Apache without recompiling?

Run
ldd httpd
As long as you installed version 1.0.1g over the same as the existing OpenSSL , and as long as you did not statically compiled libraries into httpd, it will use the newer version. Your issue is likely related to the OpenSSL.conf settings dealing with transport security layer. Check httpd.conf and DSL.conf to make sure you did not limit the TLS options.
Use sslabs.com to test the available connections, ciphers, protocols....

When you install, you used a yum update OpenSSL, or did you download OpenSSL rpm from some source and installed in such that it is now in /usr/local/openssl while the other is in /usr/....,,

Result of ldd httpd

   linux-gate.so.1 =>  (0x007cc000)
        libldap-2.4.so.2 => /lib/libldap-2.4.so.2 (0x03c16000)
        liblber-2.4.so.2 => /lib/liblber-2.4.so.2 (0x03924000)
        libssl.so.10 => /usr/lib/libssl.so.10 (0x037ce000)
        libcrypto.so.10 => /usr/lib/libcrypto.so.10 (0x03524000)
        libdl.so.2 => /lib/libdl.so.2 (0x00b79000)
        libz.so.1 => /lib/libz.so.1 (0x00bd4000)
        libm.so.6 => /lib/libm.so.6 (0x00b9d000)
        libaprutil-1.so.0 => /usr/local/apache2ssl/lib/libaprutil-1.so.0 (0x001b9000)
        libexpat.so.0 => /usr/local/apache2ssl/lib/libexpat.so.0 (0x00951000)
        libapr-1.so.0 => /usr/local/apache2ssl/lib/libapr-1.so.0 (0x00218000)
        librt.so.1 => /lib/librt.so.1 (0x00bc9000)
        libcrypt.so.1 => /lib/libcrypt.so.1 (0x00110000)
        libpthread.so.0 => /lib/libpthread.so.0 (0x00b80000)
        libc.so.6 => /lib/libc.so.6 (0x009eb000)
        libresolv.so.2 => /lib/libresolv.so.2 (0x00cf5000)
        libssl3.so => /usr/lib/libssl3.so (0x0424f000)
        libsmime3.so => /usr/lib/libsmime3.so (0x04221000)
        libnss3.so => /usr/lib/libnss3.so (0x03adb000)
        libnssutil3.so => /usr/lib/libnssutil3.so (0x03791000)
        libplds4.so => /lib/libplds4.so (0x0378a000)
        libplc4.so => /lib/libplc4.so (0x03782000)
        libnspr4.so => /lib/libnspr4.so (0x041e1000)
        libsasl2.so.2 => /usr/lib/libsasl2.so.2 (0x037af000)
        libgssapi_krb5.so.2 => /lib/libgssapi_krb5.so.2 (0x00140000)
        libkrb5.so.3 => /lib/libkrb5.so.3 (0x036af000)
        libcom_err.so.2 => /lib/libcom_err.so.2 (0x008b5000)
        libk5crypto.so.3 => /lib/libk5crypto.so.3 (0x008bb000)
        /lib/ld-linux.so.2 (0x009c9000)
        libfreebl3.so => /lib/libfreebl3.so (0x03a8a000)
        libkrb5support.so.0 => /lib/libkrb5support.so.0 (0x00909000)
        libkeyutils.so.1 => /lib/libkeyutils.so.1 (0x008e4000)
        libselinux.so.1 => /lib/libselinux.so.1 (0x00be9000)

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By default I dont think the openssl version installed over the old one because when i did 'openssl version' after the install - the version still showed 1.0.0
Rather after install I did:
mv /usr/bin/openssl /home/backup/openssltest  (backup files)
ln -s /usr/local/ssl/bin/openssl /usr/bin/openssl   (symlink)

This worked to show the new version with 'openssl version'

One option is to use ldconf to have /usr/local/SSL/lib before /usr/lib in ldconf.conf

/etc.....

The point of people choosing to go with RedHat is to maintain the version of installed applications for purpose of stability.

Your intervention means that you have to make sure to monitor the security related warnings about the external software you install and update/fix the issue.

ldconf: command not found

locate ldconf.conf
no results

I inherited this system the way it is now. Not sure why it was configured like this.

Yes, thanks this makes the most sense and is what I am doing. No need to stick with an old OS when its giving so many issues. I'll set up a new server with the newest RHEL7 and work from there.