If the Cowboys franchise tag Osa Odighizuwa, who has to go to make room?

News broke over the weekend that the Cowboys plan on franchise tagging DT Osa Odighizuwa if the two parties can't come to an agreement on an extension. Whether he's extended or tagged, he's due to make a lot more money going forward. Will the team need to cut or let go of other players at the expense of giving Osa a lump of new money?