Sawyer was set to play for Birmingham in the Alliance of American Football before getting a call from the Redskins.
The 6-3, 275-pound Sawyer was signed in May by the Kansas City Chiefs as an undrafted free agent out of South Carolina. He took part in the Chiefs' rookie minicamp before being waived.
Sawyer made 30 tackles, three sacks and five forced fumbles in 12 games at South Carolina last season.
Barnes spent time on the Redskins' practice squad last season before signing a futures contract in January. He signed with the New Orleans Saints as an undrafted free agent out of Clemson in 2015 and played in 13 games that season.
After being waived by the Saints in 2016, he went to training camp with the Seattle Seahawks in 2016 and the Tampa Bay Buccaneers in 2017, but was waived before the start of the season each time.