Try this: Create a NEW development site. Install the latest version of PrestaShop. Configure it with one item only. Activate the latest UPS module, plug in your account info and test it. If it works there, then you know it's a problem with your site - either the version of PrestaShop you are running or something else. Quite a few companies (ups, fedex, authorize.net) locked down their web services to make them more secure, and disabled old cipher suites.
Have you checked your UPS account to make sure it's still active and the API info is still valid?
I'd make a dev site if I were you, and test it there. Don't worry about installing a theme or tweaking the design. Just a raw install of PrestaShop latest version, and test your UPS account from there.