Well.
After more trouble shooting, turns out one of the servers between Verizon and Fleaflicker had blocked my IP address, which is why the anonymous proxy server worked. Verizon thinks either they made a mistake, or that I had picked up an IP address that had earlier been used by a spammer.
I unplugged my router overnight, reconnected this morning and I could get back in again without the proxy server. Woo hoo!