The Potters fell to 15th following the 3-1 loss to Aston Villa on Saturday and are only three points above the drop zone with six matches remaining.
"We are in a relegation fight and now it is about character," said Adam.
"We will see how good the character in the group is. It has got to be in question now and we have got to step up to the mark.
"We need hard work. If you don't work hard you are going to be sucked into it. We are in it.
"It is a relegation fight and we need to pick ourselves up for next week.
"There are no easy games in this league and there are none bigger than Manchester United next Sunday.
"We are at home and if we could gradually get confidence going into the game, then fine, but it is going to be a tough match for us."