Thanks for the response Jakub!

I couldn't run your command exactly because I don't use the start_tls command, I run everything over ldaps. I was able to bind anonymously using this command:

# ldapsearch -H "ldaps://test-server/" -b "uid=jharden,ou=Users,dc=example,dc=com"

I can even bind as the user using this command:

# ldapsearch -H "ldaps://test-server/" -b "uid=jharden,ou=Users,dc=example,dc=com" -D "uid=jharden,ou=Users,dc=example,dc=com" -W

I remade my CA, my server certificate, and my client certificate to make sure that I wasn't screwing something up with the certificates. Things still work with ldapsearch, but not with sssd.