If Texas is the same as Illinois then: YES

Your situation is eerily similar to what my brother went through a few months ago. He was pulled over for failure to signal and was also without insurance. This was at 1:30am on May 15. That same morning, he set-up an insurance policy and was sent a card stateing he was insured as of May 15. He took the card to court and it was dismissed. No questions asked.

Nice loop-hole I thought.