Hmm I highly doubt its the IS that is screwing you up.
Just by looking at your settings that you posted from your first pic I see a couple "mistakes". As stated earlier, the obvious one is your DOF at F4. From the angle you took the shot, obviously you were a couple feet away which leaves you with a way too narrow DOF, but you did hit the eyes anyway. Good for the eyes but bad for the rest of the dog's face because with your full frame the DOF is narrower. I would suggest again to using a higher f stop and standing farther away t give yourself more leeway to really check the sharpness.
*You cant really check sharpness of a lens or camera very well if you work with such a narrow DOF.
Next, if you still want to check on your dog I would increase your shutter speed. 125 may be the minimum to stop a dog with ADD as you put it. I would put it at a minimum of 200-250 (at least), honestly I would put it even faster if there was enough light.
If you need to raise your shutter speed, and enlarge your DOF then you will need more light obviously or raise your ISO, so mynext suggestion would be to take your pics outside in daylight where you have an abundance of light and dont need to work within the confines of such narrow parameters.
Finally you need to work on familiarizing yourself with the new AF system. Depending on your dogs movements or lack of you have to get used to what AF setting is best. This is a whole nother can of worms. If you can get him to sit still then obviously one shot focus with one spot focal point on the eye. But if not then its obviously AI servo (I recommend back button focussing) with either AF area or AF surround and picking the right AF case in the menu depending on what type of movement your dog is engaged in.
But if you really want to check sharpness - just check it on a stationary object for now lol. Learning the AF on the mark iii does take a little while and you introduce too many variables if you want to check your camera.
But from your pics my guess is that its more you than the mark iii haha.
I also highly doubt its the IS - just go outdoors and practice there a bit first. It sounds like alot of your shots are indoors and shooing at a low ISO indoors does not give you alot of room for error and requires a very steady hand and good technique.
If you do plan to shoot indoors for architecture work and dont have the time to set up a tripod I like to lean or brace my body against a wall or doorframe to stabilize myself and practice tucking in my arms into my body to create a very stable non shaking base while I take the picture. I also practce my breathing and time my shots so that I depress the shutter when I am moving the least possible amount. Think of a sniper