Hi George You have selected a subject that creates limits on your control of the results since lighting and distance in a ballet theater are your enemy. As an aside, please fill in your equipment list in your membership profile so you do not have to repeat it each time a question is asked. What lenses are you using and how close are you to the stage? First off, if you are needing to crop much and in Hi-1, noise is going to be bad and detail will be lost in post processing NR. Whatever longest lens you have, it is not long enough if you are needing to crop a lot. My go to lens in ballet is the 70-200 but that is when close enough not to have to crop. A 300 2.8 would be better but few theaters would allow it so you have to consider other means of increasing signal and decreasing noise. Do you have a particular vantage point that you are required to shoot from. I like the first balcony in a classic ornate and opulent style 250 year old theater because I can shoot down slightly, get within 20 meters of the principle dancers. Shooting up from the floor level lowers contrast due to lights just above the shooting plane, hitting the front lens element just off axis. It is the same effect when shooting into the sun with it just out of the frame, contrast goes away. I would limit ISO to about 3200 if possible, one way is lowering shutter speed and concentrating on maneuvers that are dramatic but slow in movement such as lifts. Lifts look active but they are often some of the slowest action there in on stage because balance has to be maintained to make it look fluid. Chained Adagio movements are easy to capture at 1/60 or so with VR, but you have to know the ballet well to known when slow shutter movements are coming. Much of any ballet is Allegro however so pick your spots with practiced good timing to get transition between fast Allegro movements. The slowest most graceful moves come out best in still photos and the action comes out best in video. Another issue is the direction from which the light is coming. Costumes frequently have reflective elements like metallic thread and jeweled appliques. That can create some strong spot reflections that will fool your metering and end up with underexposed images. With any of your lenses, can you fill th frame with a dancer? If not you are at a disadvantage in getting lowest noise and finest detail.
The Canon Nikon comparisons are apples and oranges and does not reflect actual use. Nikon has been setting low sharpening and moderate contrast defaults in JPG and requires the user to increase it by choice so Nikon images on DPR are always less sharp than what you really get. Canon, in recent models have tried to fight back from the trouncing they have gotten at the hands of Nikons of the same class in high ISO. They have greatly increased non-defeatable NR at higher ISO which kills detail but appears in low res magazine reviews to have less noise. They do, but also have less detail and more artifacts that are almost impossible to fix in post. You get natively cleaner files with a lot more flexibility in Nikon NEF RAW files. Compare D7000 shadow noise and artifact with any camera in their line and it is plain to see. If you are shooting JPGs only and do not blow them up, the default settings of the 7d, 5DII or 5DIII or even their 1Dx. The D600 and D800 are in another league in file quality. So do not feel as if others have an advantage, they don't. Your problem is light, too much focus on fast movements that require higher SS than the light and noise level can support, and probably not enough reach. Can you post a few images that show the typical light and distance, a finished crop and an uncropped file. That will tell us more than describing it.
The question of the D600 is a common one for good reason, its high ISO performance and DR are excellent, above all other makes but the D7000 is the best for this among DX cameras. You would get better images at higher ISO but suffer from less reach with your existing lenses if they are FX image circle. Buying reach at the long, fast end is EXPENSIVE, ask any birder how much their 400 2.8 monsters cost. A used D3s would be hard to beat for cost/noise ratio but you would still need reach because detail would be lost more by cropping a 12mpx files. Fast primes would not be the final answer due to shallow DOF that would allow little tolerance in focusing or getting two subject in the same frame in focus. You are probably going to need 2.8 for two people on about the same focal plane and maybe f/1.8 for a solo or 1/2 figure detail. So lets see some examples of what you are up against now.