Baltimore claimed Morse off waivers earlier this week before completing the deal Friday.
The 31-year-old Morse hit .226 with 13 homers and 27 RBIs in 76 games for Seattle. He missed 31 games early this season with a broken finger on his right hand. Morse spent the previous four seasons with the Nationals where he hit 31 homers with 95 RBIs both career-highs in 2011.
Avery has spent time at Double-A Bowie and Triple-A Norfolk this season. The left-hander hit a combined .259 with 70 runs scored and is 29 for 37 on stolen base attempts. He hit .223 with six steals in 32 games for the Orioles last season.