I think the cheaper and faster option is to learn how to set up the camera and post processing techniques/tools to allow you to shoot over ISO 800. This experience then can be taken to any future camera upgrade.
Based on the input, the 6D would give you nearly 2 stops of improvement, so you would shoot the 6D at nothing higher than 3200. People love to tell others to upgrade equipment, but it is often not really necessary. Glass perhaps, bodies, not so much unless you have really exploited all its capabilities.
Your 600D should have pretty clean ISO 3200 with minimal effort.
http://teamspeed.smugmug.com …3H5c/0/X3/IMG_9739-X3.jpg![]()
http://teamspeed.smugmug.com …vHqN/0/X3/IMG_6431-X3.jpg![]()
http://teamspeed.smugmug.com …fQzC/0/X3/IMG_9883-X3.jpg![]()
http://teamspeed.smugmug.com …wrgW/0/X3/IMG_2195-X3.jpg![]()
Or higher, like 4000 or 6400?
http://teamspeed.smugmug.com …8f4/0/X3/7D1_7219a-X3.jpg![]()
http://teamspeed.smugmug.com …x4H8/0/X3/IMG_2696-X3.jpg![]()
If you have some raw files at 1600 or 3200 that you don't like, try uploading to dropbox and drop a link here. I have no doubt we can at least get you up 1 stop to 1600. 
Sometimes I have been known to shoot at 12800. 













