You might be using the wrong camera mode. If you know for sure what shutter speed you need, then using shutter priority is a fine thing...but there are some key side effects of controlling the shutter rather than the aperture. For one, you might be ending up with an aperture that is too wide to keep your entire subject in focus, which could be whats causing some of your shots to be a bit out of focus (or at least partially out of focus).
You should try switching to aperture priority, and allow the camera to choose the shutter speed. Generally speaking, the camera will try its best to maintain 1/focalLength to ensure your hand-held shots have a fast enough shutter to prevent blur from camera shake. By controlling aperture, you can ensure that you use a narrow enough aperture to produce a DOF deep enough to encompass your entire subject (which may need to be as high as f/8, depending on distance and subject size).
As for noise, thats generally a side effect of the kind of photography you do. Your working with limited available light, need a certain minimum shutter speed to freeze action to an acceptable degree (unless you want motion blur, more on that in a moment), and need a narrow enough aperture to encompass your entire subject within DOF. You can compromise on aperture a bit when you really need light, and increasing the distance to your subject will help improve DOF with a fast (say f/2.8) aperture. You might just have to deal with higher ISO settings, such as 1600 or even higher, to achieve the shutter speed you need.
As for flash, well you can certainly get a powerful flash and use it to help freeze motion. With a slower shutter speed and a second/rear-curtain flash, you can also produce some nice special effects shots that keep some motion blur in place, then freeze the subject with a flash pulse at the end of the exposure. If you do use a flash, you'll need to get close enough for your flash pulse to reach and illuminate the subject. Flash guide number will help you figure out how close that needs to be. If you want to avoid using flash, then you need a very fast, very long lens. Something f/2.8 for focal lengths up to 400mm, or f/4 for 500mm and 600mm focal lengths. The first place to start would be to replace your 70-300mm lens (which are usually variable aperture f/3.5-5.6...rather slow) with a 70-200 f/2.8 lens. Keep some distance between yourself and your subject to maintain DOF, and you should be able to shoot fairy wide, or even wide open. Keep in mind that at f/2.8, your going to have a pretty thin depth of field, and that might compromise your subject focus a little...but that is usually a small tradeoff for better exposures and less noise. You will probably need to do some cropping with a 70-200mm lens regardless, so keep that in mind as well.
If you want to avoid both the need to use flash and the need to crop, then your well into supertelephoto realm. You will need either a 400mm f/2.8 lens, or a 500mm or 600mm f/4 lens. These puppies are the cream of the crop, but cost a small fortune (as in, "You could buy a car for that!" kind of fortune). These lenses also come with considerable heft, and you'll definitely need a tripod. The addition of a tripod and a gimbal type mount can be another boon, though, as it will help stabilize your shots even more, allowing you more leeway with shutter speed and ISO.
