Freelance publicist/graphic designer business
It is always recommended to form an LLC, as a DBA offers no protection of liability or separation from the company and the owner's assets. Once you have the LLC registered, you could then register a DBA (you can have multiple DBA's) under your LLC to operate under a different name. DBA's under an LLC retain the same protection, as well as the same EIN and state sales tax permits. This is something you would probably need if you have various websites where you are advertising and collecting revenue.
Once you have your LLC registered, no other business in the state of Georgia would be able to use your entity name. A business in another state may use the same entity name as yours if no entity is registered in their respective state with the same name. As far as protecting your company's name and logo design, you can always check with a trademark company to see if you would be able to trademark the name and design, making them unavailable for anyone else (at least in the United States) to use.
I hope you were able to find this answer to be helpful.