What Cliff says
Auto mode is probably a mistake.
No apparent EXIF available on your photos, but you really want to be able to select the conditions for each photo yourself. ISO, aperture and shutter speed all influence how likely a photo is to succeed. When pushed to the very limits manual mode can actually be easier on the brain than any auto or semi auto mode.
Manual mode allows you to experiment more easily.
Experiment so that you are CERTAIN where your degredations are coming from and what the limits are.
How much is camera shake.
How much is subject movement.
Is this the sort of subject where panning will help?
How much is ISO an issue (noise, camera noise correction quality reductions).
Be well acquainted with how what live-view or immediate post-view shows you, and the final results you can expect. Some cameras have superb LCDs and and images that look very good on the LCD may be far less impressive 'back home'. And for some cameras just the opposite applies - final photo quality may be superior to what the LCD shows. (In my case my A77 & A700 make/made mediocore shots look better and my Minolta 5D (6mP, still going) makes good shots look bad on the LCD).
Look at noise levels you get with a given ISO and decide what you can tolerate.
Find the best noise reduction post processing program you think you can afford and then buy the next more expnsive one if it's better.
The max ISO and best pgrogram you can manage sets the noise limits.
Translate the ISO and acceptable light level into the sort of apertures and shutter speeds you are going to need. If you have a zoom that varies aperture with focal length, know where the change points in effective aperture are and try to maximise aperture (ie minimum f.stop of course). You may only gain a 2X shutter speed by keeping the zoom in its high aperture range, but a factor of 2 can make a large difference.
Be aware of the limitations this places on your depth of field and necessary focusing accuracy and difficulty and speed of focusing. I'll indent the following narrative as it is highly relative but some people here have low attention spans and vote down chain of consciouness material - no matter how relevant.
Some personal experiences of the relative merits of large aperture lenses in non-flash low light night photography: When wandering around night markets and similar taking photos without flash I often use an 18-250 f/3.5-f/5.6 zoom.
f/3.5 lasts only from 18-24mm, f/4 from 24-40mm, f/4.5 from 40-75mm and f/5.56 from 75-250mm. The difference in light input from f/3.5 to f/5.6 is (5.6/3.5)^2 (ie stop ratio squared) = 2.6:1 or slightly more than a halving in shutter speed as aperture g=changes from minimum to maximum. That's not vast, but 1/15 to 1/30 is a great help. 1/30 to 1/60 can be extremely useful. 1/60 to 1/125 and the Ninja stabilised hand is well dealt with and you are starting to address dancer blur.
BUT when I bought a 50mm f/1.8 I thought it would vastly improve my ability to take night photos. (3.5/1.8)^2 ~= 4:1 os 4 x shutter speed. 50mm (75mm equivalent on an A700 APSC) means camera stability issues are much reduced. Rule of thumb says you need 1/50th second minimum. Ninja breathing and concentration and leaning on a post may take that own to 1/10th second and a stabilised lens may help a bit more. Pushing it you MAY get sharp shots at 0.2 s up IF the subject does not move.
BUT my night market lens of choice is now still my 18-250mm. Apart from the ability to zoom widely, it is far far far easier to rapidly focus and frame. The rather narrow depth of field of the 50 mm f/1.8 makes dealing with dynamic subjects much harder. Stop it down to f4 or so and things get better - but you may as well then use the zoom lens.
Despite the above, try a largest available aperture lens to see what it can do for you. Changing fro f/5.6 to f/1.8 (rather extreme) gives 9 times more light or 9 times higher shutter speed. 1/30th s at f/5.6 =~ 1/250th s at f/1.8. MAY make the pain of use worthwhile. Even f/4 to f/1.8 allows a 5 times shutter speed increase.
You can't choose which dance moves occur where on stage, but you can look for locations where lighting is better - especially when you are getting very poor % of good shots, selecting which ones to concentrate on based on lighting probably makes sense.
Some of your shots look substantially more brightly lit than others. The light falls more from the side in some cases. If your lights are far off the brightness may not vary much with position. If lights are close then falloff will follow inverse square law. A dancer twice as close to a light will be 4 x as brightly illuminated - 2 stops or 4 x shutter speed gain. Even 30% closer = twice as bright. Or 40% further away = half as bright.
Can you use flash at all ? - probably anathema, but maybe not.
Can you afford a D700?
Can you afford a D3s? :-).
You probably don't have any control over lighting, but if you do, some additional reflector material to increase brightness may not be at all obvious to dancers or viewers but may more than double light levels.