My question is... I'm only finding them without the DX? What does this mean and will it still suit the D5000? I'm getting confused
It's easy to become confused. The 50mm does not include the "DX" designation because it is designed for either. That lens is basically considered an "FX" lens, although as Derrel wrote; Nikon has not traditionally put that designation in the nomenclature. The traditional way to look at this is: If the lens doesn't specify anything, then it's an "FX" lens. If it says "DX" then it was designed primarily for DX sensors.
In practical terms; ANY F-mount Nikon lens will fit and work on ANY F-mount Nikon camera. Some entry-level bodies (such as yours) will not auto-focus the older AF-D lenses, and the older lenses don't communicate fully with the camera, but you can still use them if you don't mind focusing manually.
The "DX" lenses will not project an image circle large enough to cover the "FX" sized sensor, but the middle of the image will be o.k. Users of "full-sized" cameras can still use the "DX" lenses, because they can just crop off the vignetting and it's fine. Or their fancy camera will do that for them as they take the picture.
I hope this helps. If my response has created more confusion, please let me know.