GDay and welcome to the board. I am quite new here myself and am still learning but here goes...

There is no entry level doubles made in a NE chambering, NE guns are much more expensive. You can however get several new doubles for less than $5k but you can make the choice if you want one.

Spartan SPR22/Baikal double: Very basic rifles to say the least, if you want a fancy H&H, then this is not your gun. can be had in .45/70 or .30-06.
Remington news on Spartan
Spartan SPR22

Perdersolli Kodiak: Quite a nice double for the price, i havn't handled one though. Under your $5k range, and is avaliable in a few different mid-range calibres such as the versatile 9.3x74R.
Pedersoli Doubles

Merkel: The one I am saving for at the moment, very nice but about $6k AUS, which is about $4kUS.
merkel rifles
merkel prices

You are probly a few more out there...

You could look at getting a second hand double, but there arn't many for sale as the owners usually would rather part with a limb than their double! But bargains have been known to be had occasionally. Keep your eye out, you never know! Get the cash together first because you will hate to find one and not be able to buy it straight away because you don't have the cash! Happened to me twice already...damn.

Hope this helps, good shooting!