Woods was set to make $2.1 million under the restricted free agent tender he received earlier this offseason. He was credited with 33 tackles, one sack and four quarterback pressures last season, starting seven of 14 games. He joined the Cowboys in 2018 after being waived by the Tennessee Titans and outperformed expectations, starting 15 games in his first season.
At 6-foot-1, 310 pounds, Woods was more of a fit in how the Cowboys wanted their defensive tackles in former coordinator Rod Marinelli's scheme. The Cowboys selected Kentucky's Quinton Bohanna, who is 6-4, 327 pounds, in the sixth round of the NFL draft. In the third round, the Cowboys drafted defensive tackle Osa Odighizuwa, but he is projected as an interior pass-rusher.
The Cowboys also waived cornerback Saivion Smith, who spent time on the active roster and practice squad last season. With the additions of corners Kelvin Joseph (second round) and Nahshon Wright (third), Smith became expendable.
In addition to Woods and Smith, the Cowboys released center Adam Redmond, defensive tackle Walter Palmore, defensive end LaDarius Hamilton and cornerback Kemon Hall in order to reach the 90-man limit after adding their 11 draft picks and 13 undrafted free agents.