I think I got it!
The answer is:
Make a backup of the keystore file.
~ $ cp -a ~/.keystore.jks ~/baks/20140917.keystore.jks
Create a new Certificate-Signing Request file from the existing alias.
~ $ keytool -certreq -v -alias myApp2 -file 20140917_myApp2_certreq.csr -keystore ~/.keystore.jks
Go to Globalsign (or whoever your CA is), log into your account, place the order for your Code Signing Certificate Renewal.
Either upload the CSR file you created above, or open the file and copy the contents, including the "-----BEGIN NEW CERTIFICATE REQUEST-----" and "-----END NEW CERTIFICATE REQUEST-----" lines, and paste it into the textbox field for the CSR.
When you get the response email, follow the link to the page for downloading your new certificate and download the new certificate to your system.
Check that the root and intermediate CA certificates are still the same. If they are different/updated, add the new ones to your keystore and/or to your "cacerts" keystore.
~ $ keytool -importcert -v -alias gssha2root_r3 -keystore ./cacerts -file ./dls/GlobalSign/20140916_GlobalsignSHA256root_cert.cer
Install the new code-signing certificate to the same alias.
~ $ keytool -importcert -v -trustcacerts -alias myApp2 -keystore ~/.keystore.jks -file ./dls/Globalsign/OS201400000001.cer
Seems to work as expected. I am now signing my java applets with the new cert. And, the jars are verifying with the new certificate.
HTH.