Aperture Priority is likely the most popular mode on your camera. But it frequently results in a blurry subject. Oh yes I have image stabilization on my camera. That's not the problem. The background is fine. Its the subject that is blurred. You guessed it, it's Shutter speed being to low.
The theory behind of using Aperture Priority is to set the ISO and the Aperture and the camera will choose the shutter speed to result in a proper exposed photo. There are 2 problems, one is that metering may fool your camera and give an over or under exposed photo. Ok you can use Exposure Compensation to over come that. The is using Auto ISO not setting a minimum ISO to insure a sharp photo. Since I usually shoot with natural light I am subject to changes to light levels. In your camera Auto ISO setting you could set a minimum shutter speed and your camera would automatically boost the ISO to get a proper exposure. That works but what is the minimum Shutter Speed should you set? One stop higher than 1/focal length? So on my 14-150mm which on my 2x crop that means 1/300 sec as a minimum. I would set my auto iso minimum shutter speed 1/600 sec. So my camera would 1/500 or 1/640 seconds.
My choice is to set 1/125 sec when using Aperture Priority. But my real choice is use Manual.
Really it would be a semi-auto setting since I would use Auto ISO. I would choose my Aperture based on DoF and Shutter Speed based on subject. The ISO would float to give you a proper exposure. You can usually set ISO levels. Usually it is best to choose the base ISO for your camera and the upper limit based on your ISO noise tolerance. My OM-D E-M5 iii will not go above ISO 6400 in Auto. I believe this is because OM Solutions/Olympus above 6400 is an extended ISO.