You did indeed find the culprit. You do need to use a DVI cable connection. If your LCD monitor has a DVI input, I would recommend you switch regardless. You may also notice a slight improvement in overall display quality.
With the DVI connection, you can then adjust the scaling properties you need. The screen-capture George@AMD showed is for when you have an HDTV connected over HDMI, and doing 1920x1080, but getting black borders. The slider lets you "overscan" enlarge the display to correct for the borders.
In your case, you need to set your desktop resolution to something below the native resolution (same aspect ratio), and then enable GPU Scaling under My Digital Flat Panel > Properties.
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