I'll try to post one later, but even taking a macro picture of the page of a book 5-10cm directly above the page is quite blurred at the corners, only sharp in the middle. There may be a hidden setting somewhere I'm not finding, or this is normal, or there's a camera or lens fault, not sure which.
Are you using any external closeup lenses?
There are a number of observations that I take from the manual:
There is no real "macro mode" just a focus limiter.
To focus at 5 cm, you need to be in wide angle.
There are a number of settings that can affect edge focus: miniature, Toy camera and soft focus to name a few. Try resetting the camera to its defaults and try again.
If it is the case where the item you are photographing doesn't completely fit within the shallow depth of field, you can look into doing focus stacking. Though that can be quite tedious.