The Potters were relegated from the Premier League on Saturday after letting a one-goal lead slip to lose 2-1 at home to Crystal Palace.
Stoke, who have now gone 13 games without a win, have seen their 10-year stay in the top flight come to an unceremonious end, and Adam was eager to make his feelings heard in a BBC Sport interview.
"It's embarrassing because we should never have been in that situation," said the 32-year-old. "As a group you have your seven or eight who try their hardest to make it work and we failed.
"I'll be honest for the supporters, I think some players have been getting away with murder for a long time and it's difficult for supporters. It's not just one or two, I think there are four or five that could be counted.
"The biggest thing for us is we never replaced Marko [Arnautovic, who left in July 2017]. You can lose players, but when you lose players who can win you a game you are going to struggle.
"As a group of players we should hang our heads in shame because that club should never be relegated. We know what's going to happen. People will lose their jobs and we're the ones to blame for that."
Prior to the 2017-18 season, Stoke recorded three successive ninth-placed finishes under now-departed manager Mark Hughes, which was followed by 13th last term.