I happen to have a K20D, and a Sigma 70-300 (and lucky for you, an acceptable woodpecker at my feeder, like how strange is that?) so I went out and tried it with the shutter set to 1/250s, hand held in really bad light. Here's the result. I don't see how the lens would be that bad even if it needed calibrating. I'd shoot a couple more test shots, and if you can't get it to work return it. I took 23 images and checked them all to see if any were like what you have, and there were no issues in any of them, from my perspective it has to be the lens.
You're absolutely sure the camera wasn't set to MF, and the lens wasn't set to macro? (even though on mine the macro setting doesn't seem to do anything.)
Last edited by normhead; 04-08-2015 at 03:00 PM.