Canine Freestyle - Shooting action indoor w-no flash-BAD RESULTS

I photographed a canine freestyle event recently for a dog club that I belong to. I am not happy with the results. I had to shoot w/no flash and the dogs were doing a routine so moving around the room the entire time. I ended up shooting in manual mode the 2nd day because the first day, I had the ISO up to 6400 and on Shutter priority at 500 and they were too grainy. Day 2 wasn't much better at 160 speed, 1000 iso and 5.6 AV. Does anyone have any suggestions as to what I could have done better to get crisper and cleaner shots?

I was using a 5D Mark III and the lens I used was 24-105mm.

A faster lens will help. Those shots call for a 70-200mm 2.8.

But short of spending close to or above $1000.00 USD, there is no reason to not shoot at the settings you used with the setup you have. What you need to do is shoot at 6400 if you need to. Then work on your noise reduction technics in your workflow.

I normally just work with luminance in Lightroom. Run it up fairly high with conditions like this and then use masking until you get some detail back.

I do have a 70-200mm 2.8 lens but when they were moving around the room, many times they were up super close to me, they would have been chopped off, so I didn't use that lens.

Dont be afraid of high ISO on that camera. My 6D is only slightly better at high ISO than the 5D3. And I can pretty easily shoot at 25600 and have even gone to 51200 if I know I am going to convert to b/w.
While the 70-200 is the right lens because it is pretty fast, if you are willing to do some noise clean up in post, you could pretty easily get away with faster SS. The first seems to be just a motion blur issues. The second is just a poorly timed shot. If you could have gotten the shot sooner or later where the dog and the owner were both both visible, the shot looks good for focus.
It is tough, and I recently bought the 24-70 2.8 just for these same sorts of conditions.
Your camera can handle high ISO, you just have to know how to clean them up. I would rather have noisy shots that are properly focused and action frozen then blurry shots with no noise.

why wouldn't you have been shooting wide open? if light is low, why stop the lens down even more...f4 is already limiting

i would've broken the rules and used a flash...