Replying here for anyone who stumbles across this. After trying to convince me several times that the problem was due to something I had done (firewall, cookies, coding error), GoDaddy escalated the problem to their real technical support guys who immediately responded saying that there was an issue with a firewall *on my server.* They offered to move me to a different server, which I accepted, and a DNS update later, I’m back to normal.
Bottom line: You need to get past that first-line of defense at GoDaddy support to the people who know how to fix this.