I read the comments from everyone on this thread and felt compelled to respond. Like the originating poster, I too have serious arthritis problems and have had two surgeries to help fix the damage to my shoulder and back. Recoil is a serious problem for an arthritis sufferer!
Buckfever, I can totally relate to concerns about recoil from a .308. I have owned/reloaded and shot just about every caliber out there from 300 Weatherby Mag on down to .223 over the past 35 years so I have a pretty good understanding of the different levels of recoil. I spend a lot of time shooting from a bench at the range so recoil is a big issue. At one time (when I was much younger), the 308 and 30-06 were my favorite deer calibers, but not any more.
When looking for a milder recoiling caliber some years ago, I tried all the legal deer rounds in calibers from 243 to 7-08. I settled on the 25-06. It smacks deer with great authority and I have never had one walk away from it (I always use premium bullets). The recoil is very light (about 10 lbs using 100 grain bullets doing 3100-3200 fps) and is extremely accurate. I now have owned numerous models in 25-06 and every one has been a less than 1 MOA shooter with some load development using Nosler Partitions. My longest shot at big game on the prairies was a measured 320 yards....a speed goat....dead. My longest shot at a coyote was an unmentionable distance! Also dead. I once put a 100 grain Nosler Partition bullet end to end through a mule deer at aprox 200 yards across a ravine as it walked towards me.....so please don't think for a minute that the 25-06 isn't a great deer round and very accurate. Many use them for elk as well but I haven't.
I also have a liking for the 308 and have had many rifles in that caliber. It definately has considerably more recoil than the 25-06 and to me, it is a sharper or perhaps faster felt recoil, likely from the faster powder burn rate or the fact that I tended to prefer it in shorter 20" barrels. I imagine that this comment with result in some nasty responses from 308 lovers but I find the 308 does not zap 'em dead the way a 25-06 does. Agree or not, that has been my experience. Recoil is not an issue when actually hunting, heck, I don't even hear the shot let alone feel the recoil. But on the bench, it is a factor.
So, the 25-06 gets my vote for your situation without any hesitation. I also suggest you consider the new Remington rifles with the Limbsaver R-3 recoil pad....it definately works! I just bought a new 25-06 in the Remington 700 CDL version with a 24" barrel and it is one very sweet rig. Unlike many "Remington haters" I read comments from now and then here and on other forums, I have never been let down by a Remington rifle and every one of them has been an excellent shooter (with load development). And yes, I have owned Tikka, Sako, Winchester, Ruger and Weatherby rifles so I have tried many different makes.
Good luck with your situation and I hope my post helps you.