I own and shoot a 308, 30-06 and 300WM. Not much difference between the 308 and 30-06 with recoil. Big dfference with the 300WM.
No reason IMO to jump up to a 300WM if your shooting at 300 yards or less.
308 and 30-06 are great calibers for elk with a good placed shot.
If you do go up to a 300WM and you fear the recoil, then shot placement will suffer and you'd be better off with smaller caliber.
Muzzle breaks do greatly help with recoil if you don't mind it sounding like a cannon every time you shoot. It does make them loud.