Windy I would like to offer 2 prospectives to think about.
First from the hunter’s point of view.
The old 7X57 Mauser is one of the greatest cartridges ever designed. It was a wild success for many years in many countries for many types of game all the way up to elephant (in the hands of W.D.M. Bell) I have used it myself and hunting with 4 others friend that used it in the last 40 years. I have seen many many animals killed with it from 75 to 700 pounds and all were handled very well by the round. Most of my friends used 160 grain bullets. In my old 7X57 I used 175 grain bullets. The old 7X57 Mauser was originally loaded with 175 and 173 grain military FMJ and the twist was 1-9.5" They are accurate with any bullet weight up to and including 180 grain. The old models of Mauser like the 91 and 93 models were not as strong as the later M-98 so the loads are usually kept on the low-pressure side. It is also the round that was used in more Remington Rolling block rifles and their copies than we can count.
The 7-08 is the equal of the better high pressure factory 7X57 loads, so what's not to like?
You and your husband are NOT under-gunned.
Next the gunsmiths prospective;
As stated above, the old twist for the 7X57 was 1-9.5" and that what Remington uses too.
The case capacity of the 7X57 is a bit better than you 7-08, but in factory loadings the 7-08 is more powerful because there are no older weaker rifles chambered in it. Case capacity of the 7-08 is 3.32 CCs The 7X57 has a capacity of 3.68 CCs. That means if both round were loaded with the same bullet at the same pressure with the same powder, the 7X57 would edge out the 7-08 but only slightly. Bolt heads and bolt faces are the same so bolt thrust is identical in the case of both shells attaining the same pressure.
So from the gunsmith’s point of view the 260 Remington is the American version of the 6.5 Swede and the 7-08 is the American version of the 7X57.
It is true that in strong actions the older shells will drive their bullets a bit faster, but not enough faster to concern yourself.
Look at the 100 year long history of the 6.5 Swede and the 7MM Mauser and lay any concerns you may have had to rest. You have excellent rounds for hunting any animal you wish to hunt. Magnums give a bit flatter trajectories and have a bit less hang time (so less wind drift) but in my hunting experience I have not found my magnums to out perform my old 7mm Mauser or my 270, by enough of a margin to worry about. I have hunting in many states and a few other countries and have killed and seen killed many many animals. More than I can count and I am not exaggerating at all. I have been a guide in 3 states also and I have seen more deer elk and antelope killed than most.
When I tell you that your 7-08 is good for hunting anything up to and including big bull elk, I am not simply giving an opinion bases on a little experience and a lot of reading. My statement is based on a lot of actual first hand experience.
I do recommend tougher bullets that will not come apart, but I think for hunting anything that weights over 150 pounds that is good advice no matter what shell you shoot.
Use a good bullet and shoot with 100% confidence. Your rounds will be just fine.