The upcoming free agents will no longer be under contract with the Rams once the 2022 league year ends. (That's why they're free agents!)
The only two ways I can think of that they would have any effect on the 2023 cap is if they had void year(s) at the end of the contract for signing bonus distribution or if they qualified for an incentive that was considered not likely to be earned (NLTBE) in 2022.
First example:
In 2020 a player signs a 5 year $55M contract with a $10M signing bonus and $15M a year salary the first 3 years. His cap hits would look like this:
2020 $17M cap hit ($15M salary plus $2M pro-rated signing bonus)
2021 $17M cap hit ($15M salary plus $2M pro-rated signing bonus)
2022 $17M cap hit ($15M salary plus $2M pro-rated signing bonus)
2023 $2M cap hit ($0 salary plus $2M pro-rated signing bonus)
2024 $2M cap hit ($0 salary plus $2M pro-rated signing bonus)
This player would be a free agent after the 2022 season. And he would count $4M against the 2023 cap since all remaining cap money comes due upon a player's separation from the team.
Second example:
A player signed a 1 year deal for the 2022 season that has a $2M incentive for making the roster and a $1M incentive for making the Pro Bowl.
If the league office considers it likely that the player will make the Rams roster they'll label that incentive likely to be earned (LTBE) and that $2M goes against the 2022 cap. If it is not paid then the Rams get a $2M increase in their 2023 cap.
And if the league thinks the guy probably isn't Pro Bowl material they would consider that incentive NLTBE and there would be no 2022 cap hit.
The player makes the team, the Rams pay him $2M, the money is already applied to the cap, end of story.
Lo and behold - the guy makes the Pro Bowl! The Rams pay him the $1M bonus and the cap hit goes on the 2023 cap. So he generates a cap hit in 2023 even though he's not on the team.
This is the Rams' 'Dead Money' against the 2023 cap (cap hits from guys no longer on the team). I don't really know what scenario generated them ( although I'm pretty sure Whit had one or more void years on his contract.)
AlbaNY_Ram