Do you mean registered with the state for business purposes? Or registered with the USPTO as an official trademark?
T-shirt designs are automatically copyrighted once they are in fixed form. If you want more official proof of ownership, you can register the design with the US Copyright office.
While copyright registration offers benefits like proof of ownership and legal recourse, the only true 'protection' is to aggressively pursue legal action against infringement.
Putting your business name on the design doesn't offer any legal benefit. Instead, you should use the proper registration symbol (™, ®, ©).