When we purchased our home, title company listed us as "man's name and woman's name, his wife". We are not leagally married but did not speak up. Now we feel like the deed needs to be corrected to read "Joint tenants with rights of survivorship" to be correct. How do we do this? Do we have to go through title company?

You would have to file a Corrective Deed. It's simple but I would strongly recommend that you have it done by a title company to make sure it's done properly.