Cartridges don't have headspace. Rifles have headspace. Headspace is a manufacturing tolerance thing. It's the distance between the bolt face and, for a rimless bottle necked case, a spot in the chamber, where the shoulder of the case ends up. It cannot be adjusted by doing anything to the case itself.
The case length for either a 7.62 or .308 is the same. Commercial hunting ammo is loaded a bit hotter than milsurp, but not enough to bother your rifle. Prior to there being an Internet, they were the same round. Ammo was purchased by calibre and cartridge. IE. If you could find 7.62 ball and weren't hunting, you bought it. If you coudn't find ball or didn't want it, you bought .308 Win.
In your situation, wanting hunting ammo, buy a box of Winchester, a box of Remington and a box of whatever other brands you can, load 5 of each and shoot off a bench(use a bag of sand or other rest under the forestock just forward of the mag), at 100 yards, on a different target for each brand, for group only. Keep the brass. When you find the brand that gives the best accuracy, sight in with that ammo. Then practice on the pie plate. You'll avoid taking an off hand shot, if at all possible, while hunting, but knowing you can is part of having confidence in your rifle and your shooting skills.
You may well get less than 2" groups, but 2" is good enough for hunting. If you find more than one brand that will shoot 2" or less, it's a bonus.
The 165's are a good all round bullet weight for most game in North America. They'll drop a deer, a moose or a black bear with no fuss.