Last year, I tried it but the expert told me a few weeks later, she couldnt help me further and I have to use the self-filing system. Do I have to pay taxes in both states? Am I a resident of both states? Thanks for the help.
If you moved during the year, for tax purposes you're a part-year resident of each of the two states. You must file a part-year resident personal income tax return in each of the states.
Full service may not be available for your situation due to its complexity. Full service cannot file business returns, and has limits on what you can file for a personal return, (particularly if you need to issue Forms 1042-S). This Help Article provides more information on the service itself: What is TurboTax Live Full Service? - Intuit.
As far as your state situation is concerned, however, you will absolutely file a Massachusetts return because your business is based in Massachusetts. And if you earned money once you moved to California, you will file a California return. You actually will file part-year returns for both. For Massachusetts, you will claim all income you received while you were living there, and you will report any income that is "Massachusetts-sourced" after you moved. In this case, if the LLC under your control was still generating income by conducting physical business in Massachusetts that is "passed on" to you, that income will be Massachusetts sourced even if you are not physically present.
California will tax any income you earn that is taxable that is either "California Sourced" or earned once you establish residency in California regardless of its source. In this situation, you are "double-taxed". But for this portion of income, California will give you a credit for the amount of tax you pay to Massachusetts on this "double-taxed" income. And if your business has not left Massachusetts, you can expect this to be your situation going forward.
Keep in mind also that since you moved to California, the state may require you to register your LLC there as well. Check local laws for your specific requirements. California LLC organizations (taxed as sole proprietorships) must file Form 568 each year and pay an "LLC tax" (for lack of a better phrase). LLC's that have made S- or C-Corp taxing elections have similar requirements.
TurboTax is a powerful program, and our Live Full Service is a great service, but if your situation is too complex to use Full Service, your interests may be better served by a highly-qualified tax professional or CPA.
To add one item to @DanielV01 response, you may continue to be responsible to file annual reports with the MA Secretary of State, since this is the state of organization for the LLC. I recommend you call their office and explain the situation so you understand future filing responsibilities.
These filing requirements are completely separate from any income tax filings.