it depends. different states and municipal government have different rules. i'd agree with you in assuming that you'd need to get another DBA. yeah you most definitely would need to get a DBA.
Typically, if the name of the DBA is different than the name of the LLC, you will need to file that DBA with the county. From personal experience, it's called a "Fictitious Business Name Filing" in California and an "Assumed Name Filing" in Texas.
Your bank will require this for purposes of opening bank accounts in that DBA name as well as other financial matters (EINs for employees, etc).