If you want a 338 I would go with the Howa, Weatherby Vanguard, or Savage.
As far as a 444, your choices are few, TC, Marlin, and NEF (H&R), I believe that may be it, and you are going to find that you wont be able to get a new one for under $550 unless you go with an H&R.
If I have missed any I'm sure someone will let you know.
J.