What to do if it has come to my attention that a business in another state is operating a similar business with a very similar name as ours?

First, a business name is different from a trademark in Texas. A business name is not trademarked, the names/marks associated with specific goods and services are trademarked.

So if you are looking to stop someone from using your business name in another state, that doesn't sound like a trademark issue. And, your business name is probably not protected in states where you haven't used it or registered it.  

If you actually have a trademark issue---A trademark registration is valid (unless successfully contested, or lost for a range of reasons) in the state where it is registered. Additionally, your use of a mark in commerce can give you common law trademark rights, so you may have rights to your mark wherever you have used it, regardless of whether you have obtained a registration.

My initial belief is that your TM registration in Texas is the wrong place to focus.

If you have been doing business in that other state, then it's quite possible that you have common law rights in that state.

Additionally, if you have been doing business in "interstate commerce" (the other state, or, for example, general business over the internet), you might have rights you could protect on a federal level.

I would focus less on your Texas registration and more on whether you have done business in the freeloader's state or have attracted business from "interstate commerce."

Then, I would ask whether the name you're trying to protect is actually a trademark or servicemark, or just a business name that's not a mark for a specific good or service.

If justified, you would at least send a cease and desist letter, because if there is an actual trademark/servicemark issue, it's important to protect your mark against infringment. Allowing infringment will over time undermine your rights to the mark.