Can I start a LLC using a company name being used in another State?

Registration and copyright (and also trademark) are separate.
Businesses are registered state-by-state: so OH has no say over TX businesses and vice versa. So a company registerted in TX does not prevent you from registering the same business name in OH.
But sometimes business names are also protected by intellectual property law (typically trademark, not copyright). "IP" law is nationwide, so if somene has a valid trademark, then even if they are in TX, they can potentially stop you from using that trademarked name in OH. But the name you want to use would have to infringe on (be very similar to) a registered trademark for them to be able to stop you from using it, AND there would have to be some likelihood or risk of customer confusion--i.e. that a customer could think you were them or they you or vice versa. (Trademark is designed to protect and secure the identity of goods and services, and protect customers from being duped or confused.) So if you do not sell the same goods or services, so that customers might not be confused, it is unlikely that even a regstered trademark would stop you from using the name.
Copyright very rarely applies to a name, since a name generally does not meet the criteria of being an original creative work (that's what copyright protects: original creative works, like stories, films, poems, sculptures, artwork, etc.). It is possible that the LOGO is copyrighted, since a creative and original logo could be essentially a miniature piece of art--in that case, you could not use the *logo* but it would not stop you from using the name with a different look, logo, graphics treatment, etc. Copyright is very narrow--just don't use the exact original image, etc. that is copyrighted. 
As stated, trademark is much more usually the issue with business names, and is broader than copyright--even similar (not identical) names can be barred if too much like a trademarked name. The U.S. Patent Trademark Office (PTO) has a very good website where you can search to see if a name is trademarked: search there, to see if trademark is an issue.