Personally, rather than getting a whole new camera, I'd get a new lens, a faster one. Just being curious, what lens/lenses are you using with the Canon Rebel? I can see that your 55-200mm is quite slow, it has an f/4 aperture at it's widest, and an f/5.6 at it's longest zoom.
Let me explain something so you can understand what I'm talking about. A fast lens is a lens that has a large aperture. Large apertures are equal to smaller f numbers, I'm pretty sure you've seen something in your lens like this: f/4-5.6. That is the f number or aperture. Fast lenses can let more light in and consequently lead to faster shutter speed which can get you sharper photos. One thing to consider is that the depth of field (DOF) is reduced, this is the amount of stuff that will be in focus. Let me explain.
Just a small note, this is probably inaccurate, but it's just to demonstrate. If you focus in the same place, when you have your lens at f/5.6, your DOF is 2 meters front to back of what will be in focus, but if it's in f/2.8, only 50cm will be in focus.
Let me explain differently, probably more clearly. Imagine you have a large ruler starting at the end of your lens and extending forever. Imagine you focus at the 50cm mark. When you have your lens at f/5.6, you will have from the 25cm mark up to the 80cm mark in focus, but if you have it at f/2.8, you will only have from the 40cm mark up to the 65cm mark in focus, and if you have it at f/1.4, you will only have from the 45cm mark up to the 55cm mark in focus. This is imprecise, but it's a general idea of how aperture affects your DOF.
Another aspect to consider is getting a lens with IS (or in Nikon lenses, this is known as VR). This is basically a motor (I think) inside the lens that compensates for the movement of your hands to get sharper shots. In my experience, IS or VR is VERY helpful, it has allowed me to get sharp shots in really dark environments.
Now let's recap. To get a faster shutter speed, you need a faster lens, that means, a lens with a smaller f/ number. This reduces your DOF, so be careful with moving objects. Another thing worth mentioning is that lenses with faster apertures are usually bigger, heavier and more expensive. Also IS or VR will help a lot, but this also adds up to the price and weight.
If you still get blurred pictures with a fast lens, you might need to upgrade your camera to be able to bump up the ISO a little higher without introducing so much noise.
In case you don't understand ISO or noise, let me explain. ISO is the sensitivity of the camera when you take a picture. ISO starts in most cameras, at 100 ISO, and it doubles itself in this fashion: 200 ISO, 400 ISO, 800 ISO, 1600 ISO, 3200 ISO, and so on and so forth. Each increment of these is one light stop, and it helps so you can get faster shutter speeds. Shutter speeds increment in 1 light stop as well, so if you start with a shutter speed of 1/8 of a second and 100 ISO, which will probably get you a blurred shot, when you increase the ISO to 200, the shutter speed increases to 1/15 of a second. So when you get your ISO to 400, your shutter speed is 1/30, and with ISO 800 it will be 1/60, and with ISO 1600 it will be 1/120. At this point, you'll probably get a sharp picture as long as you have steady hands (or IS), but here is where noise comes in. When you increase ISO, small colored dots start appearing, especially in dark areas of the picture. This is known as noise. Better cameras often have less noise at higher ISOs, and cameras with larger sensors will give you better results. Cameras with large sensors are known as full-frame cameras.
_________________Gear: Canon SX20 IS, Canon Rebel T3i, Canon EF-S 18-55mm, Canon EF-S 55-250mm f/4-5.6 IS II, Canon EF 50mm f/1.8 II
Wishlist: Canon EF 100-400mm f/4-5.6L
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