In the 1st photo, the baby's eyes are not in focus, but the hand and front of the comforter are in focus, so the lens is front focusing. It was taken at ISO 1000, f/4 (max f/4), 1/250 sec, 105mm. Good settings. Very nice photo if it were sharp. In the 2nd shot, the model's eyes are not in focus nor are the letters on the t-shirt. It was taken at ISO 100, f/4 (max f/4), 1/10 sec, 65mm. I can't tell if the lens is front focusing, but the shutter speed is low and it could be camera shake. In the 3rd shot, the models eyes are soft, but the tree in front of the eyes is sharp, so again, front focusing. It was taken at ISO 100, f/4.5 (max f/4), 1/320 sec, 120mm. Good settings. Very nice shot BTW, beautiful model, great pose, excellent composition. You are obviously a very good photographer. Was it the same lens in all 3 shots? Is it a 3rd party lens?
What to do? All I can tell you is what I would do. I'd start by checking the lens. Set up a still scene, something with a full page of sharp text at about a 40 degree angle (high in the back, low towards the front) and something with texture. You want the text, like a magazine page, at an angle so you can see what is in focus and what is not. Tripod, remote release (or timer), ISO 100, wide open aperture, shutter speed to get a good exposure. AF-S, spot focus (note where your focus point is), matrix metering. Take 3 or 4 shots using the viewfinder and 3 or 4 shots in Live View. The viewfinder uses phase detect AF and Live View uses contrast detect AF. Phase detect is faster, but not as accurate. Contrast detect is accurate, but not as fast. My bet is you will see the AF is fine in LV, but front focusing with the viewfinder.
As far as sluggishness in p mode. That's probably a settings issue. Check and see if you are set to focus before releasing the shutter.