I have a 336 in 35 remington and a 7600 in 308. Like both although they are not my primary go to rifles. The 7600 gets little respect but I have found it to be as accurate as any M700 I've owned.

That being said, I looked heavily at the 7mm-08 in a bolt action not that long ago. I live in rural Maine and ammo selection was an issue. bascially there was no place to get any without either a long drive or mail order. Turned me off of that cartridge. If you can get the ammo, I think it would be a great choice.

I think there are a number of rifles out there that fit your needs. Remington M7s, Savage, Ruger etc. I'd look into something like that in 308. Would be significantly different that your 06 but will have less recoil. Just my opinion.