Don't think so, the cache is really for content. Something like the POST request wouldn't go through the cache tables since it's not cacheable content.

One thing I'm wondering - I don't know anything about the UPS system... do they require you to register an IP address? Is your site at the same domain? One obvious thing would be that your new server is at a different IP than the old one; if your IP was logged at the time you registered your UPS service then that would make sense as to why it's rejecting the new site.

Another thought along those lines would be a hardware firewall. Usually GoDaddy (depending on your setup) has a firewall in place, and depending on their rules, could be blocking the IP incorrectly. I don't see why since that request is going through standard HTTP / HTTPS ports but again, it might just be a site wrongly listed as a spammer.

One final thing to consider would be contacting GoDaddy and asking them to take a look at any error logs for your site, if they can. Assuming they are competently hosting your site (always a crapshoot with them) there should be some logs that might hold some key information as to why these things aren't happening.