Gotta love those muzzle brakes....the one on my 16" 308 Win. Savage Scout survived two trips to the range (until I forgot to pull down my ear muffs before the fateful shot!). It now wears a thread protector.
Felt recoil has a lot to do with rifle weight and fit. I've noticed that my 375 h&h has less "felt" recoil than my .338 WM. The 375 is more of a push, the 338 a sharper (quicker) recoil (338 & 300 WMs are both a sharper recoil, it seems, but there may be more at play than bore dia.??
I'd guess your 185 gr 338 will be a very flat shooting round; a lot flatter than the 180 in the '06 and the '06 is near the top of bullet weight where the 338 is just beginning.
Regards,
hps