The Seahawks confirmed the deal, which saw them send the Jets first-round NFL Draft picks in 2021 and 2022, a 2021 third rounder and safety Bradley McDougald in exchange for star safety Adams and a 2022 pick in the fourth round.
Adams, in an interview published Friday by the New York Daily News, critized Gase in a move to inspire a trade that proved successful.
"I don't feel like he's the right leader for this organization to reach the Promised Land," Adams told the newspaper.
"As a leader, what really bothers me is that he doesn't have a relationship with everybody in the building."
Adams seeks a new contract and began pushing for a trade in June because he was unhappy with a lack of progress in talks.
"At the end of the day, (Gase) doesn't address the team," Adams told the News.
"If there's a problem in the locker room, he lets another coach address the team. If we're playing (badly) and we're losing, he doesn't address the entire team as a group at halftime. He'll walk out of the locker room and let another coach handle it."
Adams, the sixth overall pick in the 2017 NFL Draft, has made 273 tackles, 25 pass deflections, two interpcetions and forced six fumbles in 46 games for the Jets over three seasons, all as a starter.
The Seahawks had the fifth-worst pass defense in the NFL last season.