Conditions in the first pic look quite challenging, as there's not a huge amount of light available for the camera to focus in. Also, your camera has probably chosen a fairly wide aperture to let more light in which means the depth of field (ie the area that's considered to be sharp and in focus) is quite small.
In the first photo, the dog's nose looks to be sharp but not the eyes - so you can see the depth of field is quite narrow. Now, if you wanted to make sure the eyes are in focus you would aim at the eyes, wait for the red dot and beep, and then click. But you're wanting to re-compose which makes things harder.
First of all, recomposing will change the focus plane. Imagine you hold your arm outstretched so it's parallel to the ground, palm forwards as if you're touching a wall. Now if you move your arm up or down it's no longer touching the wall, right? That's similar to what happens when you recompose - the focus plane is moving, and the eye is not the same distance away as the nose so you get the wrong bit in focus.
There are several ways around this... all of them a compromise to some extent.
1) Choose a composition where the eye is in the centre. Or at least near enough. For this shot
I moved so I was above the dog and aimed between her eyes. I then cleaned up the composition by cropping all the wasted space around her.Here's a shot of the same dog from a different angle.
2) You could use an external flash to provide more light, which will allow you to stop your aperture down which will give you more depth of field (more areas of the photo in apparent focus). This way you don't need to be quite as accurate - you'd get more depth of field at f/6.3 than you would at f/3.5 for example.
3) You can select a different focus point on your camera. Your success with this will depend on various factors; for example, on Rebel cameras only the centre focus point is a cross-focus point, meaning it's more accurate and has a better chance of locking on focus. Using the outer points tends to be slightly less accurate and works best with more light as well as a more contrasty subject area. Also, the focus points may not be in a very useful position for the shot, and by the time you've messed around the dog has moved.
4) Focus and recompose, then compensate slightly. This takes practice and is often hit or miss, but sometimes you can focus and get an idea about how much you need to move the camera forward or back to compensate for the focal plane moving.
Photographing animals can be a very frustrating experience... but we live in an age where the cost of taking a photograph is almost nil, and your patience will eventually pay off. I tend to work on the principle that 1 in 20 photos will be good enough that I'd want to show someone or print them out, so take a hundred and you might get 5-10 decent in-focus shots where the animal's looking in the right direction, not blinking, etc.
The second shot you posted isn't too bad. I guess it depends on your tolerance for how sharp the eyes have to be. I'd be quite happy with it, anyway. When a dog's walking towards you it will have moved a few inches between you focusing and then pressing the shutter button. You could increase the f/stop, as mentioned earlier, or try out the AI Servo mode on your camera. Some of them are bound to be keepers.
Good luck!
