I would suggest a Weatherby Ultralight or a Tikka Superlight. The Weatherby's are both 30-06 and I usually shoot 150's or 165's, if you think the recoil might be too much I believe they are made in 308, 270 and 7-08. My SL is a 308 and doesn't kick much. For a scope I like the Leupold VX-3 2.5-8x36, some of the guys on here like a Leupold fixed 6x. Good luck!
I'm going to second the Weatherby. I'm only 64 years old, pretty good sized and in great shape, so I don't feel like I have any physical limitations. However, I happen to like the Wby ULW, mine's in .257. Not the very lightest but for a long rifle, balances well and the stock design is the best. It directs recoil away from your face and puts your eye where it belongs on a scoped rifle. It has it's detractors, to be sure but I suspect that few have actually fired one. A couple years ago, I had the same rifle in 7mm Wby mag and it was still pleasant to fire, even from the bench. I often wish I had the 7mm back. Accuracy with the ones I've owned was exceptional.
The other thing I'd suggest, if you feel you have physical limitations is to take the money you'd spend on a doctor (or a new rifle), hire a knowledgable trainer and get a gym membership. You might surprise yourself. I rehabilitated my right shoulder at the gym, slowly and carefully, after being told it needed surgery.
