Seems like a DNS problem on the server.

A test could be to TEMPORARILY add a hosts file entry:
Code:
56.0.34.43  production.shippingapis.com   #temporary for testing only, Oct 2015; should be removed ASAP
If the problems stop immediately after putting that in the hosts file but re-occur again after removing it, then that strongly suggests that the server's ability to do name resolution is broken.

Do remember to remove it from the hosts file after you're done testing, in case USPS changes the IP address in the future.