Reasons to Suspect
Low appetite, low stomach acid, trouble with fats, low red blood cell zinc levels
Oysters are amazing. In general, animal sources of zinc have a favorable zinc:copper ratio, while plant sources have more copper. Liver and pumpkin seeds are the exceptions.
For supplements, zinc picolinate is best absorbed (Barrie, et al, 1987), but also more easily excreted. The ideal form seems to be something that dissolves well in water, but isn't too tightly bound. Still searching on what that translates to.