I think cutting it down would get you new growth. It would get cut down by the winter here in 8b naturally but When I lived in Hawaii, I never cut it down and never noticed it, so I think it is fine either way you decide..

I think that in south Texas you won't need to cut it down. I've had several of them in my yard for years, and I have never cut them down. They even survived the 2004 6" Christmas snow without any protection. And living so far south, I think that you should do just fine and can continue to use your lemongrass year round. You might want to check out lemongrass. Good luck!