The 26-year-old midfielder joined Liverpool last season from Blackpool, but found himself out of favour with new Anfield manager Brendan Rodgers.
He has signed a four-year deal at the Britannia Stadium.
"It's no secret that we've been looking to bring more goals to the squad and Charlie is a fantastic signing in that respect," Stoke manager Tony Pulis told the club's website.
"Not only will his creativity and his eye for delivering a pass be a great asset to us, he scored 12 goals for Blackpool in the Premier League two seasons ago."
Adam said: "This club's on the up and it's great to be a part of it.
"I hope I can help them to improve upon the success which has been achieved over the past few years."
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