You shouldn't need another EIN, because it is granted on a federal level, so you should be covered in both states, and 48 more, for that matter!

To register your name, you should contact the local government agencies in both states. It may be different there, but where I'm located, you can get all the information from the city/county clerk in the area your business operates out of.
You may also want to register a federal trademark, if you want to protect your name from liability, but that is a matter of personal preference. I hope I helped, if not ask again!
Best of luck to you!