The larger sensor, even over a APS-C will give shallower depth of field which some users just don't like. It also means AF accuracy is more critical, being off just a little affects the image noticibly.
At low ISO's, do not expect a noticeably better image from FF, that's not where the difference shows up.
I always recommend doing a AFMA on a new camera or lens. Using the rear LCD to determine sharpness can be misleading unless you put the camera on a tripod and use magnification.
If you have a close subject, using one shot can cause focus errors just do to tiny movements of your body.
Its a complex subject, and in order to help you, you need to post a link to a full size image and supply your camera settings. Otherwise, we assume that its a focus error just because it often is the issue.