Check the USPTO database and your state's Secretary of State office before filing your LLC name. State approval doesn't protect you nationwide; another business in a different state can still use the same name. A federal trademark is the only way to secure exclusive rights across all states.
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Before you register your LLC, make sure to check that the LLC name is available. To do this, you'll have to check with the USPTO office, which is the Trademark Office, to make sure that nobody already has a trademark under the name that you desire. Once you do that, you'll have to check with the Secretary of State's office in the state that you want to register your LLC in to make sure that nobody else has an LLC under the same name. If neither the USPTO or your Secretary of State's office show that the name is taken already, then you you can proceed with filing the articles with your state. Obviously, it doesn't guarantee that the state's going to accept the name, but this does strongly increase the likelihood that your name gets accepted. Also keep in mind that just because the state does approve your LLC does not mean that your name is protected. Another business in a separate state can register an LLC using the same name as yours. The only real way to protect your business's name is to get a trademark.