A lawyer will be able to advise you based on Texas law who was there first and has the right to use the name.
This really is a good issue you brought up.
You can put it off, but you will need to keep it in the back of your mind, if they had the name first, they could force you to change it. How would that effect your business? These are things to consider.
I don't know if the TX counties have these business records online or not, but you could look into it and investigate it further. Then if you were registered first, you would have better standing.
Let me know what you think.