It's delusional to think he will be cheap, much less sign for the vet minimum which is under a million dollars.
First, it's only speculation at this point that the Jets may release Adams.
If they did, he may not clear waivers. Unlikely he would be claimed by a team out of the playoff race. More likely to be claimed by a team in the playoff race.
If not claimed off of waivers Adams would be free to sign with any team he chooses. Would likely sign with a playoff contender for the remainder of 2024.
Most believe if Adams were to be released by the Jets and become an unrestricted free agent, he would command upwards of $13 Million for 2025.