Bread is too moist in a humid climate
Flour-free breads are, in my experience, very hydroscopic: they absorb water from the air even faster than regular bread.  However, since the bread you're using is presliced, there's a simple solution: store the bread in the freezer.  Just separate the slices (so that you can pull them apart after it's frozen), and put it in a ziplock freezer bag.

This does mean that you'll need to toast, or at least thaw, the bread before using it. However, I've found that freezing bread results in less of a "stale" flavor than refrigerating it.