New hunter, what rifle?

[Badflies]

Hello all. I am a middle age dude who hasn't shot anything larger than varmit. I am looking into a first rifle and I'm overwhelmed by the selection. I would like to get something soon and get some time in at the range before the fall. I'm looking for deer and elk. I have lots of practice chucking slugs out of a lightweight 12 gauge and I'm a big guy so I don't think I would be recoil sensitive. I don't want a cannon though as I go through a fair bit of ammo at the range and some of it isn't cheap.
So I'm thinking $1000ish tops with a scope. Popular candidates look like a savage axis, Mossberg patriot or Remington 783 in 270, 6.5, or 308. I would love a Winchester 70 or Remington 700 but they look a little pricy.
What would you guys recommend? What should I avoid?