I have tried everything in my power to fix this, and I've gotten nowhere.
I went into Windows firewall and made sure Hamachi-2.exe is allowed in every case. Nope, still probing.
I went into my router settings and made sure port 12975 is open. Nope, still probing.
I even changed the DNS settings to use Google DNS instead of my ISP's default DNS. Nope, still probing.
It seems like the only other option is a proxy server. This is possible, though. I've found that when I start up Google Chrome, instead of instantaneously loading webpages like it used to, it now hangs for a few seconds on "Resolving Proxy" before loading the webpages. Could it be that AT&T somehow changed my proxy server settings? I had not messed with any router settings before Hamachi stopped working, so it could not have been my doing. I have no reason to connect to a proxy server. How can I find out what kind of proxy server Hamachi is trying to connect through?
This is so nerve-wracking...