Suns coach Monty Williams has been a significant supporter of Shamet since coaching him as an assistant with the 76ers.
The Nets now own the No. 27, 29, 44, 49 and 59 picks in Thursday's draft.
Shamet, 24, averaged 9.3 points per game and shot 38.7% from 3-point range for the Nets last season.
Carter, 25, played 12 minutes per game last season and averaged 4.1 points and 1.2 assists.