While I am a fan of the .270WIN, for game of Elk size and under at 300 yards or less, there is not a whole lot of performance difference between it and the 30-06.
With the modern bullets available now, either will work just fine ..... the only real differences are that there are more different .308" bullets available than .277" varieties .....but how many different bullets do you really need? The -06 has heavier bullets available ...... and the .270 will have a couple of offerings lighter than the .30cal has ..... and a .277" bullet generally be more efficient than one of equal weight in .308" diameter ..... that and hunting ammo for the .270 will generally have lighter bullets than similar -06 ammo, so the .270 will recoil less, all else being equal .....
I'd second the idea of hitting the pawn shops/consignment racks at the LGS to find a hunting rifle- deer caliber rifles don't get shot very much, and are likely in good shape in the wear department ..... most folks just don't shoot their deer rifle much, as it is not as inexpensive or comfortable to plink all day with an -06 or .270WIN bolt gun as it is with, say, an AR or an SKS ......
TheGolden Rule of Tool Use: "If you don't know what you are doing, DON'T."