AF-S 55-200mm DX and the AF-S 55-300mm Dx are the best options. Realize though that there are two different 55-200mm lenses; one with VR (Vibration Reduction) and one without. The 55-300mm lens has VR.
Unfortunately, the least expensive lens; the 55-200mm non VR version is still $200.
You can get it refurbished though for around $100.
http://www.adorama.com/NK55200DXR.html
And you can get the 55-200mm VR refurbished for only a few dollars more - so this might be the better value:
http://www.adorama.com/NK55200VRR.html
And finally, the 55-300mm refurbished is still in your price range:
http://www.adorama.com/NK55300VRR.html
Thing about refurbished - they are still new lenses, but they cannot be sold as new. They might be a demo unit, lenses that did not initially meet factory specs off the assembly line, or a return. They are checked out by the factory and have a 90 day warranty. At least that is a better deal than used.
I would only buy factory refurbished though, not store refurbished. All of the refurbished lenses at Adorama and B&H are factory refurbished.
The 55-200mm in my view has a slightly better lens construction as it has internal focusing. The 55-300mm on the other hand does not. This means the barrel rotates as the lens is focused. If you want to use a polarizer filter on this lens, each time the lens focuses, the filter will change position.
And there is not a significant difference in focal length between 200mm and 300mm, so you are trading a slightly better lens for slightly longer optics.
Look at this webpage for a comparison between 200mm and 300mm.
http://www.althephoto.com/concepts/lenses.php
And don't forget the 70-300mm, which is superior to the other two. It is more expensive, but you can get one refurbished from Adorama for probably less than you can buy the 55-300.
http://www.adorama.com/NK70300AFVRR.html
The 70-300mm lens is a step up from either of the Dx zooms. I bought mine refurbished, and have had no problems with it.