If something is engorged, it has taken in so much liquid that it looks swollen, tight, and over-full. The liquid is often blood. Doctors use the word for tissue, a vein, or an organ that is packed too full. You will also hear it of a tick or a leech after feeding, and sometimes of a river after heavy rain. The picture is physical: stretched from the inside. People do sometimes stretch it to mean “having eaten far too much,” but that is a looser, less careful use. It is a science and medical word more than a playground word; it sounds precise, not poetic. You can feel the related verb underneath it: something has been filled to the point of swelling.