GWS declared on Wednesday afternoon that Shaw would become a Giant after an agreement had been reached with Collingwood, though the paperwork is not expected to be finalised until Thursday.
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The Giants and Magpies have been locked in a long stand-off over the deal, with both clubs placing a higher value on their player.
A premiership player with the Magpies, Shaw had two years remaining on his contract, while Adams was out of contract after two seasons with the Giants.
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GWS football manager Graeme Allan said Shaw had "plenty of good football ahead of him" after signing a five-year deal to join the club.
"He’ll be able to provide leadership and support for our exciting young group and they will benefit enormously from his presence," Allan said.
"He has been a great Collingwood favourite for a number of years and we think somebody of his undoubted quality will be a tremendous asset for the club."
Adams was selected at pick No.13 in the 2011 NAB AFL Draft, making him the fourth first-round selection to join the Magpies in 12 months.
Collingwood will also take picks No.6 and No.10 to this year's draft.
Adams said he was relieved to finally have his move confirmed after a three-week wait.
"The word was that it would happen but the clock was counting down and until it is done there is always doubt," he said.
"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."
As Shaw is 28 years old and closer to the end of his career than the beginning, GWS pushed for a swap of later picks to sweeten their deal.
Collingwood, however, believed it should receive a draft 'kicker' given Shaw would join the Giants with a strong track record at the Magpies.
In the end, a compromise was reached with the Magpies to pay a portion of Shaw's salary next year.
The delay in confirming the deal was also caused by the Giants' focus on securing a deal for pick No.2 in the 2013 NAB AFL Draft from Melbourne.
Absolutely wrapped to be at @GWSGiants excited for the challenge, thanks to everyone for there support it's been totally overwhelming— Heath Shaw (@heathshaw39) October 23, 2013