Any excuse will do me for buying a new deer rifle. However, if you still like the 7400, try some Remington Managed-Recoil 30-06 loads. You might be surprised.
You might also be interested in jumping into reloading. Recoil reduction was the major reason why I got into it. By using a powder like Hodgdon H4895, and using 150 grain bullets, I was able to find a 30-06 load that was still deadly on deer, but was light enough that my 9 yr old son could shoot comfortably. I do pretty much the same thing now with my 308 WIN, 35 Whelen, and 7.62X54R Mosin Nagants. It's amazing how much recoil you can take out of a load by reducing it 8% or so off the max. Those first 30-06 loads I cooked up for Mooseboy were still able to cycle my Remington 742 and M1 Garand with no problem.
BTW: if you get a 308 WIN, do not expect less recoil. There may be more. For one thing, you are shooting a gas-operated semi-auto. A good part of the recoil goes into cycling the action-- about 15%. The recoil on 308 WIN and 30-06 are roughly the same. If you buy a bolt-action in 308 WIN, the felt recoil will be actually more. Additionally, the 7400 is a fairly heavy gun. The big difference between a 30-06 and 308Win is the length of the action. Shorter receiver means less metal. Less metal means less weight. Less weight means more recoil. This is why I warn everyone away from short magnum actions for deer. Unless you're trying to lose weight to deal with the rigors of higher altitude, a little extra weight on your rifle means a lot less recoil.
My Girlfriend, KYHillChick, wanted to get into shooting. I bought her a 30-06. She found the 30-06 recoil to be acceptable in a big heavy M1 Garand, but when I put her in a bolt-action that was 2 lbs lighter, she developed a flinch. It is a rifle that will never see a life beyond the bench, so weight was no problem. My solution was to put 7 lbs of lead in the stock. Now she has a 15 lb gun that has negligible recoil. Everyone who shoots it laughs, because what all the lead does is cause almost no initial recoil. There is a noticeable interval after the blast and then the shoulder gets moved back firmly a few milliseconds later.
If you are not into reloading, I can also suggest a couple of chamberings you may like. 7mm08 is a good one. The 300 Savage is also a good choice-- it's loads are lighter than the 308 WIN, but it still delivers a decent punch on a deer at 200 yards. I like 300 Savage so much that I took my Savage 99 in 308 WIN and downloaded the round to match the 300 Savage in performance. That is now my GOTO rifle for whitetail out of a treestand. 243 WIN is another round that folks seem to love, but I think of it more as a varmint round.
I prefer the 6.5X55. Several years ago I did a study on all the commonly used rifle cartridges looking for the best performance for the related recoil. The .280 Remington was in there, the .300 Savage was in near the top. However, the 6.5X55 Swede was IT. The .260 Remington was also in there, as was the .257 Roberts. I have shot all these at one time or the other. All of these will kill deer under normal hunting conditions. Remember this study was based on a strict ratio of something like TKO (Taylor Knock-out) vs. felt recoil. Some of these cartridges I mentioned have low TKO, but also phenomenally low felt recoil. You may be able to take a higher recoil value and therefore be able to put a bit more punch downrange.
EDIT: (I went back and looked and found the results. I was usuing a derivative of the TKO formula called the Beker KOV)
How did the 30-06 fare in the study? As I remember, it was always a contender, no matter how I set up the study. The fact that it is regarded as the universally general all-purpose cartridge is borne out. There is always something that will do better than the '06 given a specific set of criteria, but no cartridge does it all so well.
EDIT: Here's a link to the stuff from my weblog in [url=http://blackholecoffeehouse.blogspot.com/2005_02_01_archive.html]February, 2005.
As to the elk and bear part, the '06 will do well on all North American game. Some of the cartridges I've mentioned won't. Unless you plan on selling the 7400, that'll make a good elk/bear gun, and then concentrate on a low-recoil deer gun.