You can't sign a player to a contract without renegotiating their current contract. This is by design, players in the final year of their contract must be renegotiated in the current season or left to become a free agent to help guarantee they get market value.
You can try making some cuts to free up the cap room you'll need to extend your better players, the offers should be able to be submitted and the cuts will be processed before the free agent signings, you just don't have a guarantee that the offers will all go through.