Help with indoor action lighting

I've been having a little trouble with shooting indoors. I have a 20D and use a 2.8 70-200 lense. I shoot horse shows, mostly outside but I have a gig every year that requires me to shoot indoors. I have a big problem with using a flash because it will turn the horse's eye's white as well as surrounding skin, and without a flash they are pretty dark. When I try to make them better in photoshop the colors just turn out terrible. Is there any good way to shoot in this type of condition. Is there a certain shutter sync or flash or some type of equiptment that might help me?

Try some negative FEC. That will at least help for the overexposed skin. I suspect the eyes are reflecting the flash, so there will be no avoiding that. Use a brush in Photoshop to make 'em black.
Some examples with EXIF will also help getting more detailed answers.

How dark is it inside? How close can you get? 135mm f/2 or 85mm f/1.8 might be a good choice. I would try to avoid flash if possible? If not, move the flash off the camera?