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Edit. RESOLVED! I forgot to restart the webpage after I bound the cert.
Showing up properly on public IP and deployed to users properly.
The workaround is to contact GoDaddy and have them reissue your organization's certificate. During the certificate setup process, you must select a SHA-1 codesign certificate instead of SHA-2. The option to select SHA-1 will only be available if you certificate validity does not extend to 2016 (see below), so make sure they understand your end goal is to recreate your SHA-2 certificate as SHA-1, so they know to sell you a cert with the correct validity period.
I traded my SHA-2 cert for a SHA-1 today, and GoDaddy's Java Code Signing instructions worked perfectly.
GoDaddy informed me Keytool may have trouble importing a certificate response chain generated from their SHA-2 (2048 length) codesign certificate. I withhold judgment of Keytool since it imports SHA-2 certs fine when the GoDaddy's root SHA1 cert is lopped from the pem file per @mogsie's answer.
GoDaddy goes with SHA-2 automatically when it grants codesign certificates that will extend into 2017 because Microsoft will not accept less than SHA-2 beginning January 1, 2016, so if you're in the market for a SHA-1 certificate, it will have short-term validity.
The issue might go away with a Java Keytool update (I was working with 1.6), or if GoDaddy's Sha256withRSA self-signed certificate becomes widely trusted.
The answer, as mentioned by Waterbear, is to have your GoDaddy cert reissued or rekeyed by GoDaddy using SHA-1. The reason is that GoDaddy has two CA servers: Class 2 CA which is used for signing SHA-1 certificates, and G2 CA which is used for signing SHA-2 certificates. While the older Class 2 CA is trusted by the Java Truststore (and thus SHA-1 certificates are trusted), the newer G2 CA is not, so its SHA-2certificates are not trusted unless you manually install its root certificate (which defeats the purpose of buying a cert in the first place). Hopefully GoDaddy's G2 CA becomes trusted by the Java Truststore soon (Before 2016!), but until that happens a GoDaddy SHA-2 cert is no better than a self-signed cert.