If the Defendant is a corporation doing business in Texas, and it does not have a registered agent in Texas, then the Secretary of State is their agent for service of process. In that case, service is effective when it's delivered to the Secretary of State, even if the mail from the Secretary of State ultimately not picked up.
In order to get the court to enter the default judgment, you'll have to satisfy the judge that the defendant was doing business in Texas, and does not have a registered agent in Texas.
Effecting service on a defendant under the Texas long-arm statute, which is what you are describing, is highly technical. The judge will require strict adherence to the applicable procedural rules before he or she grants you a default judgment in the event the defendant fails to answer timely. Accordingly, if you don't have a lawyer, I stongly suggest that you hire one. Even if you get a default judgment, that is only a piece of paper. Converting that piece of paper into money requires a formal legal education, as well as training and experience in collecting judgments. This is particularly true where, as here, the defendant resides is a state different from the one in which the court renders judgment.