Yeah, I think there is more than one thing going on here to contribute to your bad photos.
We need more info or examples posted with exif data still intact to really help. Otherwise, we're all just guessing.
I'd start out by switching to P mode, instead of any of the green box or lower modes, as they make it difficult to tweak settings sometimes. Change your AF mode to AI Servo, center focus point only, and change your drive speed to Hi Speed. Hold the shutter button down halfway, keep your AF point on your subject, and let the AF track it. The squeeze the shutter fully whenever you want to take a photo. Practice releasing to the half-press position to keep the AF tracking. Try that out for action shots and see if things improve. I suspect they will as long as you're shooting in sufficient light. This all assumes your shooting technique is good as well. Many people simply hold SLR cameras wrong, or stab at the shutter button, etc, and the contributes to blurry images due to camera shake sometimes.
RAW vs JPEG is a totally different issue and generally relates to correcting exposure in post processing. It won't make up for bad technique or let you fix out of focus images.