Your California LLC whether formed on San Diego or anywhere else in California can do business as (dba) in any other city on California as you would file the required (dba) in the county the LLC is doing business in.
So on your example the LLC formed on California would be doing business in Los Angeles so the LLC file a fictitious business name statement in Los Angeles. It would be xyz LLC doing business as abc on Los Angeles county.
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Unless you indicate to the public that the business is an LLC, then prudent thing to do is to file a dba statement in Los Angeles based upon your facts above. You do not need to register in San Diego if you will not be doing business there.
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Yes, a fictitious business name without "LLC" at the end is appropriate.
It is not clear whether you currently are using the fictitious business name in SD (the fact that the LLC's address at the time of formation was in SD is irrelevant). If so, then the FBN (DBA) should be filed there. In any event, it should be filed in LA once you move there.
This information does not constitute legal advice and does not establish an attorney-client relationship.
The are large legal, business and tax consequences to registering as a dba or forming an LLC for your business (or for that matter any other corporate form). An attorney can discuss these differences and recommend the best option for you depending on the character of your business and your individual needs.
Principle place of business refers to where you direct and manage business activity. However if you register in San Diego you can still do business in Los Angeles. If you have no physical address yet, you can put something down and supplement thereafter. Please note, the City of Los Angeles may have a separate tax to fulfill in order to properly be registered to do business. An attorney can look at your specific situation and advise you on the appropriate steps for your move and any further dba/LLC considerations.