Ah! Selecting the entire region, like you suggested, in addition to selecting the layer, did the trick. *headdesk* I can't believe I missed something so simple - I swear I've read through the GIMP documentation so many times and somehow missed the obvious.
Thanks so much for your help!
And David - thank you, as well. That's the method I've been using, and it works like a pro. Now I can do both methods.