As the owner of about a dozen .308s ( including a Sauer 200 Lux ) and a dozen .30-06s, I would say:
* If you are hunting deer, get a .308 with 22-inch barrel and keep the weight down.
* If you are going to hunt elk, big boar, and big game at perhaps long range, and I could only have one, I would take a 24-inch barrel lighter weight .30-06 like my Rem M700 BDL SS.
* If you want to make one rifle do double duty on roe deer, fox, coyotes, etc, get a 7mm-08 or 7x57 or .270, so you can shoot better 100-gr and 110-gr bullets at reasonable speeds.
Since you already have a Sauer 202, borrow a barrel and try both cartridges. A .308 with 165-gr bullets is pretty potent, and the monolithics like the 165-gr Barnes TSX and TTSX give it some extra ooomp a big game at ranges over 200 yards.