What are your camera settings, specifically shutter speed? You could be seeing motion blur from the camera, the subject, or both. I assume you're indoors for the portraits, so are you using a flash?
When you recompose your shots, make sure the camera does not re-focus before you take the picture. I have been on a Back Button AF binge of late, and I have found it really helpful to prevent the camera from refocusing when I depress the shutter. I'm not very good at holding the shutter halfway and moving the camera at the same time.
Always use a tripod for precise focus, especially with portraits, is good advice. I will chime in and add using the 2 second timer on the shutter release can help reduce camera motion, too.
If you have not invested in a sturdy, robust tripod, then you should consider doing so. I would advise buying tripods and heads rated at least 30 pounds, or more. Manufacturer load ratings frequently tend to be over-optimistic. Most of the tripod load ratings are probably done with the center column fully retracted, but they don't tell you that. I use tripods without center columns, or at least very short ones.
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"Enjoying photography since 1972."
