Yes, you need to sign them again. The certificate itself will tell it's own expire date, and the certificate is deployed along with your package. It's an entirely new certificate you've got (even if the issuers often refer to it as a "renewal").
Installers and other verification software will usually not use the Internet to check the validity of the certificate. Instead, they will check the expire date in your certificate file (which is packed into the signed JAR file), and check the validity of the certificate by checking against the computes' built-in list of issuer certificates (CA). The only time the Internet is used in this process is to download a revocation list - a database of certificates revoked before their expiry date - but this will usually not be done in realtime, but on a scheduled basis.