Moore missed some practice time during training camp with a hamstring injury. He went on injured reserve last season with the same issue, playing 11 games and starting five.
He must miss at least three weeks before returning to the active roster.
The Texans signed Thomas to a one-year, $2 million contract this offseason before releasing him earlier this week.
Thomas, 29, is a former undrafted free agent from South Carolina State who initially signed with the Packers in 2015. He spent three seasons with Green Bay and three with Dallas.
Last season with the Cowboys, Thomas made 48 tackles in 15 games. He saw action on 409 defensive snaps and 170 on special teams.