Drew was selected in the fifth round by the Ravens this year. The former East Carolina star was battling an ankle injury late in training camp and will be the fourth tight end on Miami's roster.
Martin had 31 catches for 450 yards and three touchdowns last season for Miami.
The Dolphins also finalized their practice squad Monday, which includes: defensive end Ryan Baker, cornerback Will Billingsley, linebackers J.D. Folsom and Danny Lansanah, offensive linemen Nate Garner and Mark Lewis, and receivers James Robinson and Julius Pruitt.