According to the formulas I have seen, the Cowboys are getting comp picks for Chauncey Golston and Jourdan Lewis. They aren't getting a comp pick for DeMarcus Lawrence due to the value of the Dante Fowler signing.
So it all depends on what Cooks' deal is for compared to what Dallas gave guys like Parris Campbell, Miles Sanders, etc.. Right now the Cowboys lose a comp pick for any player they sign for over $3.5 million per season. If Cooks is getting more from the Saints than Dallas gave Campbell, for instance, it could lead to a 6th or 7th round comp pick.
That all ends after the draft, when teams can sign anybody and the comp pick formula goes out the window. That's probably why we won't see Dallas sign anyone else until then.