While researching old newspapers, I found this short, front-page article in the Breckenridge (Tex.) Gazette published on November 6, 1929. It mentions an effort (which eventually failed) to centralize, and modernize the Texas prison system. As the article explains, there were three competing proposals, one of which nominated Sugar Land as the location for the new centralized prison.
An article at the Texas State Library and Archives explains what happened. The Legislature passed a funding bill, which Governor Dan Moody refused to sign. It provided enough funds to upgrade existing facilities, but not a penny to build a modern, centralized prison. Click here to read the article. (Click here to read more on that era of the Texas prison system's history.)
Central Units 1 & 2 were constructed as a result of the Legislature's 1930 appropriation for upgrades. I'll have to research the exact dates, but I'm pretty certain they were built in 1930 & 1933.