This answer has two components. First, a majority of states have naming requirements for LLCs and regular corporations that would prevent you from using a name that was already taken by a registered corporate entity in the state. From the information you provided me, it sounds as if the other LLC that you are speaking about is not in your state. If this is so, you should be able to reserve your desired name.
On the trademark issue, I assume you are speaking about federal, not state, trademark protection. Given that the Texas LLC is no longer in business, there are three ways to analyze this issue.
First, if the Texas LLC registered the name with the Texas State Secretary, never filed a federal trademark application, and did not use a particular brand in its marketing efforts, the firm’s name does not have common law trademark protection or federal statutory trademark protection. If this is so, you can use the name you want to reserve in economic activities inside and outside of Texas.
Second, if the Texas LLC registered the name with the Texas State Secretary, never filed a federal trademark application, but used a particular brand in its marketing efforts, the firm could have common law trademark rights that extend only to the areas that it did business in. If this is so, you can use the name you want to reserve but I would encourage you to refrain from doing business in the Texas area.
Finally, if the Texas LLC registered the name with the Texas State Secretary and filed a federal trademark application. The Texas firm owns the mark. This does not necessarily mean, though, that you cannot also use the mark. You mentioned that the Texas LLC is no longer in business. This is significant because if the firm failed to file a declaration of continued use or a renewal application with the United States Patent and Trademark Office (USPTO), or you choose to initiate a cancellation proceeding with the USPTO based on the fact that the Texas LLC is no longer using the mark, you could appropriate the trademark for your own purposes. I hope this helps.
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