I can't speak for Texas as I'm not a resident. But I can tell you about Missouri...
Missouri accepts NR certification, meaning that they require nothing more than an Application, Background Check, and a copy of your National Registry card. I had my MO EMT license within 2 weeks of mailing them those copies.
Now, still regarding Missouri, I got by with simply faxing them an application and a copy of my NR Certification #. They stated they handle background checks (Plus I just had to renew my EMT license through them a few months back). So its only about a 10 day turn around for my Medic Licensure
4-6 weeks sounds like a "Blanket Term". Maybe they are throwing that number out there so that they aren't flooded with angry callers if they don't mail out in one week. Or they could be one of the states that only process applicants in "Batches" (Wait until they have X number of applicants then issue licenses).
Keep in mind that I know nothing of Texas's practices, just throwing the ideas out there