The likes of Robin van Persie, Cesc Fabregas, Samir Nasri and Alex Song have all been sold in recent seasons because their deals were due to shortly expire and the club did not want to lose them on a free transfer.
England winger Theo Walcott is also in a similar situation at the moment. His contract at the Emirates Stadium will end next summer and so far talks over fresh terms have stalled.
Adams, who spent his entire career with the Gunners, believes that a change in policy is needed if the club are to challenge for the major trophies again.
"I never got into the last year of my contract when I was at the club. David Dein, and the people who were running the club, always made sure I had two or three years to run [on my deal]," he told talkSPORT. "They shouldn't have sold Robin van Persie because you need to keep hold of your best players. They made a mistake there, they should have wrapped Robin's contract up earlier.
"It's not about finance, they are going to miss Robin's goals. If you then drop out of the top four then the money [they would have spent on van Persie] would have paid for itself.
"I can see the fans are frustrated, however I do agree with Arsenal's principles. No one player is bigger than the club.
"They need a structure and a formation but they need to do a little bit of work towards it. If Jack Wilshere plays well are they going to lose him? Arsenal need to start winning things."
Arsenal have won a trophy since defeating Manchester United penalties in the FA Cup final back in 2005.