Shaw's career as a Pie ended on 173 games, with the running half-back admitting a change was necessary for him amid rumours of a rift between player and coach Nathan Buckley.
He had already indicated his desire to move north and join the Giants for their third season in the AFL onwards, but it took over a week to have the deal finalised.
"Heath brought passion and heart to everything he did at Collingwood. I hope he’s proud of what he achieved here. He should be," Buckley said.
"Heath will always be a Collingwood premiership player and hopefully the opportunity to build something almost from the ground up with the Giants will see him recapture his best."
Midfielder Adams played 31 games for the Giants, but returns to his home state after being drafted from the Geelong Falcons with pick 13 in 2011.
"Taylor is a player a lot of clubs thought highly of when GWS picked him initially, and a player a lot of clubs were interested in when it became apparent this year that, for family reasons, he wanted to return to Victoria," Collingwood recruiting manager Derek Hine said.
"He comes to us with a deserved reputation for being strong in the contest, a talented stoppage player who has great endurance. Importantly, we feel there is a lot of development in his game still to emerge.
"All things being equal, Taylor should have a long and distinguished career in black and white."
Adams added: "The word was that it would happen but the clock was counting down and until it is done there is always doubt."
"The Giants put two years into me and helped me enormously and with some really close friends at the club there will always be a part of me that wishes the club well.
"It was really a decision based on a desire to be closer to my family and friends and the idea of playing for a big club in Melbourne. I met the coaches and saw the facilities and knew then I wanted to be a part of it. I couldn’t be happier."
GWS also announced they have signed delisted Geelong defender Josh Hunt to a one-year deal.