Palmer should not be here next season, he will be 35. Bannan will be 36 turning 37 midway through the season.
As OP said, sentiment should be going out the window now. It doesn't matter if they've had promotion or a fan favourite, it should be entirely, are they good enough to offer something to the squad. Personally, I keep Bannan as he offers something to the team. But we don't need a squad of players the other side of 35.
The only role I want to see Palmer in next season with us, is as over age U21 player-coach. That gives him his playing contract and a way into coaching which he clearly wants. Its also a modern football club role. Look at Man Utd, they have had Tom Huddlestone, Paul McShane and now Tommy Rowe all doing that job