After retrying 2 or 3 times, I realized what was happening. My street name is Boxford, and some idiot programmer decided that anything with the string "box" in it was a P.O. box. Brilliant!
Anyway, instead of taking hours to explain this problem to some customer service rep who doesn't deal with website problems, I went to the nearest Verizon store and got my phone there. I ended up paying $20 more for an accessory package (car charger, etc.) that was cheaper online, but I'll be writing Verizon to complain about that.