For your price point, keep the 30-06 Savage and get a decent scope for it. 30-06 with the right load works for everything in North America. Get a Nightforce SHV scope and move it to a new rifle in a few years. The NXS would be a hundred over your price point (unless you are LE or mil/vet).
You could get a pre-fit barrel for you Tikka in 30-06 for $400 to $1000 (Proof Research carbon) as well, just depends on how fancy you want to get. That would give more for your money than a whole new rifle. You could still get an SHV if you got a $400 barrel.
A Winchester Model 70 featherweight SS in 30-06 can be had from Midway USA for under $1000. If you just want a nicer rifle this is an excellent option and a gun with soul, making it worth keeping for a lifetime.
Out of your price with anything other than a cheap scope, but if weight is a concern and you like to tinker it is hard to beat a Kimber Montana for $1250 from EuroOptic. If weight is not a big deal for you, get the Winchester.
For a rifle that needs nothing out of the box a Sako Model 85 cannot be beat; but short of a used one this is out of your price range. I'm trying to sell mine. i would be happy to give more information. But, if it where me, I would get the Winchester for true CRF (but the Sako over any push feed).