If you don't like the recoil of a 30-06 and it bothers you that much, there is no way that you will like the .270 either as I highly doubt you will be able to tell any difference. I'd get a .243, 25-06, or possibly a 7mm-08 if you are recoil sensitive.
That said, I don't think the Savage Axis is the greatest budget rifle out there, but it's not horrible either. A friend has one in 25-06 he bought for his son to use and for a hunting rifle it does just fine. The trigger is heavy, and the stock feels cheap but the rest of the gun is solid. It's killed many a deer in the past two years from several different people shooting it. I've killed one with it myself. It's just flat out fun to shoot. Now that said, I think the Savage 10/110 is a 10x better rifle and imo it's worth paying the little bit extra for. If you can't pay the extra, I think I'd also lean towards the Ruger American over the Axis. However, I'd pick the Axis over any of Remington's lower end rifles.