The first thing I can recommend is to try reloading. Before you sink money into a new rifle, look at continuing to shoot 30-06 with a slightly reduced load. You can knock a good deal of recoil out by reducing the load 5-8% off the MAX and suffer very little reduction in performance. This is what got me into reloading in the first place. For one caliber, you can easily start reloading on the cheap. PM me if you're interested, and I'll tell you how to get set up.
Second. Look at the fit of your rifles. If the stocks are not fitting you correctly, you may be getting knocked around unnecessarily. 30-06 is not that much of a kicker, especially on a semi-auto like the 7400. It may be time to visit a gunsmith and see about getting that stock adjusted a little and maybe having a pad installed. A good recoil pad will take a lot of sharpness out of the recoil.
Third: What kind of load are you using? For instance, 180 grain bullets kick a lot more than 150 grainers-- a lot more. That's one of the tricks that these Reduced Recoil loads do. I think the Remingtons go all the way down to 130 grain.
The 308 WIN is going to kick about as much as the 30-06. Performance of the 308 WIN is similar to the '06, but the rifle itself is usually going to be lighter by several ounces (short action vs. long action). 7mm-08 is a good choice, but there again, you're going to have a short-action receiver. I'm not saying anything bad about either choice, just that there's offsets at play here. The whole point is to reduce recoil. Instead of 308WIN, I'd look at 300 Savage-- very similar cartridge, but much lighter recoil than 308 WIN or '06.
One other thing I can suggest is to add some weight to the stock. You'd be surprised what that can do. KYHillChick was getting thumped by her 30-06, so I added some lead bullets into the hollow plastic gunstock and it brought the recoil down tremendously.
Before making any major moves, download a copy of POINTBLANK software (it's free) and use the recoil calculator to figure out what you've got and use it to plan on where you're going. As a for instance, your current rifle may be putting out 20 ft lbs of recoil at around 13 ft/sec. If you just go to a 150 grain bullet it may go down to 17 ft lbs and 11 ft/sec. Adding a half-lb of lead to the butt stock might bring it down to 15 ft/lbs and 10 ft/sec. . You've knocked 25% off the force of the recoil and dramatically reduced the recoil velocity and it hasn't cost you anything.