Carp spent time with Boston and Texas last season, when he missed more than a month because of a broken right foot. He hit .175 with no homers and 13 RBIs in 59 games.
Texas acquired the 28-year-old on a waiver claim from Boston in early August, then designated him for assignment less than a month later.
The left-handed hitter has a career .254 batting average, 27 homers and 123 RBIs in 887 at-bats across six seasons with three clubs.
Washington also agreed Tuesday to a minor league deal with catcher Steven Lerud, who has 15 at-bats in the majors, all with Philadelphia.