I don't know that the Saints didn't try to sign Woods to the PS, but that wouldn't change the present circumstances with Woods. Teams are free to sign any player off any practice squad to their roster, so him being on the Saints PS wouldn't prevent him from playing for the Bucs. IIRC, the Steelers signed him right after the rosters were trimmed to 53. Reggie Jones was waived on the same day as Woods.
IMO, it says a lot about the Saints scouting department that a team has a guy that couldn't make the active roster in their starting rotation.