It sounds like you are a sole proprietor. If that's the case, then you often will do business under an assumed name, "John Smith, doing business as 'Art by Smith'", for example. You don't have to, though -- if you don't have a d/b/a, you don't have a d/b/a.
If you want to form an LLC, the LLC will have its own name which will be up to you. You can do it as "John Smith, LLC" if you want, but if you're going to create a new LLC, you should at least think about naming it something like "Art by Smith, LLC."
There is a potential problem when your personal name is also the name of the LLC -- it can confuse people into thinking that they're dealing with you personally, instead of dealing with the LLC. And if that happens, then those people may be able to sue you personally, instead of just the LLC. Since you're creating an LLC to protect yourself against those suits, it's probably best to name it something different. Best to get the advice of a local lawyer on that.
You don't need the DBA when you get the LLC. That LLC is the business name. You will also need to get a Business Tax Number. That's what the bank will really be interested in.
As Chris F. mentioned, and I can tell you from experience, anyplace you display your business name always include the LLC at the end. Folks can and will claim they did not know you were an LLC and will sue you personally.